<PAGE>



                                                                   EXHIBIT 10.56

                                                                  EXECUTION COPY




                                    THIRD AMENDMENT AND WAIVER dated as of July
                           19, 2001 (this "Amendment"), to the Amended and
                           Restated Credit Agreement, dated as of February 14,
                           2001, as amended and restated as of March 30, 2001
                           (the "Credit Agreement"), as amended, among ALAMOSA
                           HOLDINGS, INC. ("Superholdings"), ALAMOSA (DELAWARE),
                           INC. ("Alamosa Delaware"), ALAMOSA HOLDINGS, LLC (the
                           "Borrower" and, together with Superholdings and
                           Alamosa Delaware, the "Alamosa Parties"), the Lenders
                           party thereto (the "Lenders"), EXPORT DEVELOPMENT
                           CORPORATION, as Co-Documentation Agent, FIRST UNION
                           NATIONAL BANK, as Documentation Agent, TORONTO
                           DOMINION (TEXAS), INC., as Syndication Agent, and
                           CITICORP USA, INC., as Administrative Agent and
                           Collateral Agent (the "Agent"); and Amendment to the
                           First Amendment.

                  WHEREAS the Alamosa Parties have requested that certain
provisions of the Credit Agreement be amended or waived, in the manner provided
for in this Amendment, and the Required Lenders, Majority Term Lenders (as
defined below) and each Term Lender party hereto are willing to agree to such
amendments and waivers on the terms and conditions hereinafter set forth. For
the purposes of this Amendment, the term "Majority Term Lenders" means, at any
time, Term Lenders having Term Loans and unused Term Commitments representing
more than 50% of the sum of the total outstanding Term Loans and unused Term
Commitments at such time.


                  NOW, THEREFORE, the parties hereto hereby agree as follows:

                  1. Defined Terms. Capitalized terms used and not defined
herein shall have the meanings given to them in the Credit Agreement, as amended
hereby.

                  2.  Amendment to Credit Agreement.  Section 5.16 of the Credit
Agreement is hereby amended and restated in its entirety as follows:

                    "SECTION 5.16. Delivery of Revised Business Plans. No later
               than October 1, 2001, Alamosa Delaware shall deliver to the
               Lenders a 10-year business plan of each of Alamosa Delaware and
               the Borrower, in form and substance satisfactory to the Required
               Lenders."

                  3. Waiver. Each of the undersigned Lenders hereby expressly
waives, solely with respect to the borrowing on or prior to August 14, 2001 by
the Borrower from Term Lenders that have not extended their Term Commitments
pursuant to Section 4 of this Amendment of Term Loans in an aggregate principal
amount equal to the lesser of $50,000,000 and the aggregate principal amount of
Term Commitments due to expire on August 14, 2001 that have not been extended by
the Term Lenders pursuant to this Amendment (i) any Default or Event of Default
existing under the Credit Agreement as of the date hereof and (ii) the
limitation on borrowing set forth in Section 4 of the Second Amendment dated as
of June 7, 2001, among the Alamosa Parties, the Lenders party thereto and the
Agent.

<PAGE>

                                                                               2

                  4. Extension of Term Commitments. (a) Each of the Term Lenders
having a Term Commitment under the Credit Agreement that executes a separate
signature block appearing on its signature page hereto set forth therein for
such purpose agrees that the scheduled expiration of a portion of its Term
Commitment on the six month anniversary of the Original Effective Date, as
provided in Section 2.09(b) of the Credit Agreement, shall be automatically
extended without any further action on the part of any Person to December 31,
2001.

                  (b) Notwithstanding the provisions of Section 2.02 of the
Credit Agreement, the Majority Term Lenders hereby agree that (i) the Borrower
may make a drawing of Term Loans pursuant to the waiver set forth in Section 3
above on a non-pro rata basis from the Term Lenders that have not extended their
Term Commitments pursuant to this Section (which Term Loans shall be made by
each non-extending Term Lender on a basis proportionate to its share of the
non-extended Term Commitments) and (ii) any Borrowing, subsequent to a Borrowing
pursuant to clause (i), by the Borrower of Term Loans up to the aggregate
principal amount of the Term Commitments extended pursuant to this Section shall
be drawn on a non-pro rata basis from the Term Lenders that have extended their
Term Commitments pursuant to this Section (which Term Loans shall be made by
each extending Term Lender on a basis proportionate to its share of the extended
Term Commitments).

                  5. Effect of Amendments and Waiver. Except as expressly set
forth herein, this Amendment shall not by implication or otherwise limit,
impair, constitute a waiver of, or otherwise affect the rights or remedies of
the Lenders, the Collateral Agent, the Syndication Agent, the Co-Documentation
Agent, the Documentation Agent or the Administrative Agent under the Credit
Agreement, as amended by this Amendment, or any other Loan Document, and shall
not alter, modify, amend or in any way affect any of the terms, conditions,
obligations, covenants or agreements contained in the Credit Agreement, as
amended by this Amendment, or any other Loan Document, all of which are ratified
and affirmed in all respects and shall continue in full force and effect.
Nothing herein shall be deemed to entitle the Alamosa Parties to a consent to,
or a waiver, amendment, modification or other change of, any of the terms,
conditions, obligations, covenants or agreements contained in the Credit
Agreement, as amended by this Amendment, or any other Loan Document in similar
or different circumstances. This Amendment shall apply and be effective only
with respect to the provisions of the Credit Agreement set forth herein. After
the date hereof, any reference to the Credit Agreement shall mean the Credit
Agreement as modified hereby. This Amendment shall constitute a "Loan Document"
for all purposes of the Credit Agreement and the other Loan Documents. Each of
the Additional Security Agreements (as defined below) shall constitute a
"Security Document" for all purposes of the Credit Agreement and the other Loan
Documents.

                  6.  Representations and Warranties.  Each of the Alamosa
Parties hereby represents and warrants to the Agent and the Lenders as of the
date hereof as follows:

                  (a)  No Default or Event of Default has occurred and is
continuing.

                  (b) The execution, delivery and performance by the Alamosa
Parties of this Amendment have been duly authorized by all necessary corporate
and other action and do not and will not require any registration with, consent
or approval of, notice to or action by, any person (including any governmental
agency) in order to be effective and

<PAGE>

                                                                               3

enforceable. The Credit Agreement as amended by this Amendment constitutes the
legal, valid and binding obligation of each of the Alamosa Parties, enforceable
against it in accordance with its terms, subject to applicable bankruptcy,
insolvency, reorganization, moratorium or other laws affecting creditors' rights
generally and subject to general principles of equity, regardless of whether
considered in a proceeding in equity or at law.

                  (c) All representations and warranties of the Alamosa Parties
contained in Article III of the Credit Agreement (other than representations or
warranties expressly made only on and as of the Restatement Effective Date) are
true and correct in all material respects as of the date hereof.

                  7. Effectiveness. (a) Other than with respect to (x) the
waiver set forth in Section 3, which shall become effective immediately upon
satisfaction of the conditions set forth in paragraph (b) below and (y) the
amendment set forth in Section 4(a), which shall become effective with respect
to each Term Lender executing a separate signature block appearing on its
signature page hereto upon receipt by the Agent of counterparts hereof duly
executed and delivered by the Alamosa Parties and such Term Lender, this
Amendment shall become effective (the "Amendment Effective Date") only upon
satisfaction of the following conditions prior to 7:00 p.m., Eastern Standard
Time, on July 20, 2001:

                  (i) the Agent shall have received counterparts hereof duly
         executed and delivered by the Alamosa Parties, the Required Lenders and
         the Majority Term Lenders; and

                  (ii) the Borrower shall have paid an amendment fee to (x) each
         Lender without a Term Commitment that has delivered an executed
         counterpart of this Amendment to the Agent by the Amendment Effective
         Date and (y) each Lender with a Term Commitment that has delivered an
         executed counterpart of this Amendment, including a separate executed
         signature block on its signature page pursuant to which such Term
         Lender has agreed to extend its Term Commitment pursuant to Section 4
         of this Amendment, to the Agent by the Amendment Effective Date, in
         each case equal to .125% of the aggregate amount of such Lender's Term
         Loans, Revolving Exposure and unutilized Commitments at such time (such
         payment to be made by wire transfer of immediately available funds to
         the Agent for the respective accounts of such Lenders).

                  (b) The waiver set forth in Section 3 shall become effective
only upon satisfaction of the following conditions:

                  (i) the Agent shall have received counterparts hereof duly
         executed and delivered by the Alamosa Parties, the Required Lenders and
         Majority Term Lenders; and

                  (ii) (A) the Alamosa Parties and the Collateral Agent shall
         have entered into a security agreement and a blocked account agreement
         and such other instruments or documents as the Collateral Agent may
         reasonably request (the "Additional Security Agreements"), each in form
         reasonably satisfactory to the Collateral Agent; (B) pursuant to the
         Additional Security Agreements, the Borrower shall, simultaneously with
         the borrowing pursuant to the waiver set forth in Section 3 above,
         deposit the cash proceeds from the Term Loans

<PAGE>

                                                                               4

         borrowed pursuant to the waiver set forth in Section 3 above into an
         account with the Collateral Agent (or an Affiliate thereof designated
         by the Collateral Agent) governed by the blocked account agreement,
         which cash shall secure the Obligations (other than the Roberts
         Obligations, the Southwest Obligations and the WOW Obligations (each
         as defined in the Security Agreement)); and (C) the Additional
         Security Agreements shall provide that, so long as no Default or Event
         of Default shall have occurred and be continuing, upon presentment by
         the Borrower to the Collateral Agent from time to time after December
         31, 2001 of paid invoices in form reasonably satisfactory to the
         Collateral Agent from third party vendors evidencing purchases on
         commercially reasonable terms by the Borrower (or a Subsidiary Loan
         Party) of additional Collateral to be used in the Wireless
         Telecommunications Business of the Borrower and its subsidiaries in
         which the Collateral Agent has a first priority perfected security
         interest under the Security Agreement, the Borrower may substitute
         such additional Collateral for cash on deposit in the blocked account
         and cash in an amount equal to such invoices shall be released to the
         Borrower from the blocked account (and the Collateral Agent shall be
         authorized and directed by the Lenders to take such action as may be
         necessary or advisable to release such funds).

                  8. Expenses. The Alamosa Parties, jointly and severally, agree
to reimburse the Agent for its out_of_pocket expenses in connection with this
Amendment, including the reasonable fees, charges and disbursements of Cravath,
Swaine & Moore, counsel for the Agent.

                  9.  GOVERNING LAW; COUNTERPARTS.  (a) THIS AMENDMENT AND THE
RIGHTS AND OBLIGATIONS OF THE PARTIES HERETO SHALL BE GOVERNED BY AND CONSTRUED
IN ACCORDANCE WITH THE LAWS OF THE STATE OF NEW YORK.

                  (b) This Amendment may be executed in any number of
counterparts and by different parties hereto on separate counterparts, each of
which when so executed and delivered shall be deemed an original, but all of
which together shall constitute a single instrument. Delivery of an executed
counterpart of a signature page of this Amendment by facsimile transmission
shall be as effective as delivery of a manually executed counterpart hereof.


<PAGE>


                                                                               5

                  IN WITNESS WHEREOF, the parties hereto have caused this
Amendment to be duly executed and delivered by their respective proper and duly
authorized officers as of the day and year first above written.


                              ALAMOSA HOLDINGS, INC.,

                                  By /s/ Kendall W. Cowan
                                     -------------------------------
                                     Name: Kendall W. Cowan
                                     Title:  Chief Financial Officer


                              ALAMOSA (DELAWARE), INC.,

                                  By /s/ Kendall W. Cowan
                                     -------------------------------
                                     Name: Kendall W. Cowan
                                     Title:  Chief Financial Officer


                              ALAMOSA HOLDINGS, LLC,

                                  By /s/ Kendall W. Cowan
                                     -------------------------------
                                     Name: Kendall W. Cowan
                                     Title:  Chief Financial Officer


                              CITICORP USA, INC., individually and as
                              Administrative Agent,


                                  By /s/ John P. Judge
                                     -------------------------------
                                     Name: John P. Judge
                                     Title: Vice President and Director






<PAGE>


                                                               SIGNATURE PAGE to
                                            THIRD AMENDMENT, CONSENT AND WAIVER,
                                                       dated as of July 19, 2001
                                                        to ALAMOSA HOLDINGS, LLC
                                           AMENDED AND RESTATED CREDIT AGREEMENT




                         To approve the Third Amendment, Consent and Waiver:


                         Name of Institution:

                                By
                                  -----------------------
                                  Name:
                                  Title:




                         To approve the extension of the above-named
                         institution's Term Commitment pursuant to
                         Section 4(a) of the Amendment:


                         By /s/ John P. Judge
                           ------------------------------
                           Name: John P. Judge
                           Title: Vice President and Director




<PAGE>

                                                                               7

                                                              SIGNATURE PAGE to
                                            THIRD AMENDMENT, CONSENT AND WAIVER,
                                                       dated as of July 19, 2001
                                                        to ALAMOSA HOLDINGS, LLC
                                           AMENDED AND RESTATED CREDIT AGREEMENT



                         To approve the Third Amendment, Consent and Waiver:


                         Name of Institution: COBANK, ACB
                                             ----------------------------------

                                           By /s/ Anita Youngblut
                                              ----------------------------
                                              Name:   Anita Youngblut
                                              Title:  Vice President




                         To approve the extension of the above-named
                         institution's Term Commitment pursuant to
                         Section 4(a) of the Amendment:


                         By /s/ Anita Youngblut
                            -------------------------------------
                            Name:   Anita Youngblut
                            Title:  Vice President


<PAGE>

                                                                               8

                                                               SIGNATURE PAGE to
                                            THIRD AMENDMENT, CONSENT AND WAIVER,
                                                       dated as of July 19, 2001
                                                        to ALAMOSA HOLDINGS, LLC
                                           AMENDED AND RESTATED CREDIT AGREEMENT




                          To approve the Third Amendment, Consent and Waiver:


                          Name of Institution: THE BANK OF NOVA SCOTIA
                                              -----------------------------

                                            By  /s/ Stephen C. Levi
                                              -------------------------
                                              Name: Stephen C. Levi
                                              Title: Authorized Signatory




                          To approve the extension of the above-named
                          institution's Term Commitment pursuant to
                          Section 4(a) of the Amendment:


                          By /s/ Stephen C. Levi
                             -----------------------------
                             Name: Stephen C. Levi
                             Title: Authorized Signatory









<PAGE>

                                                                               9

                                                               SIGNATURE PAGE to
                                            THIRD AMENDMENT, CONSENT AND WAIVER,
                                                       dated as of July 19, 2001
                                                        to ALAMOSA HOLDINGS, LLC
                                           AMENDED AND RESTATED CREDIT AGREEMENT




                            To approve the Third Amendment, Consent and Waiver:


                            Name of Institution: TORONTO DOMINION (TEXAS), INC.
                                                --------------------------------

                                              By /s/ Carolyn R. Faeth
                                                 --------------------------
                                                 Name:  Carolyn R. Faeth
                                                 Title:  Vice President




                            To approve the extension of the above-named
                            institution's Term Commitment pursuant to
                            Section 4(a) of the Amendment:


                            By /s/ Carolyn R. Faeth
                              -------------------------------
                               Name:  Carolyn R. Faeth
                               Title:  Vice President









<PAGE>

                                                                              10

                                                               SIGNATURE PAGE to
                                            THIRD AMENDMENT, CONSENT AND WAIVER,
                                                       dated as of July 19, 2001
                                                        to ALAMOSA HOLDINGS, LLC
                                           AMENDED AND RESTATED CREDIT AGREEMENT




                 To approve the Third Amendment, Consent and Waiver:


                 Name of Institution: FIRST UNION NATIONAL BANK
                                     ----------------------------

                                   By /s/ Stephen G. Locke
                                     -----------------------
                                     Name:   Stephen G. Locke
                                     Title:  Assistant Vice President




                 To approve the extension of the above-named institution's
                 Term Commitment pursuant to Section 4(a) of the Amendment:


                 By /s/ Stephen G. Locke
                    ----------------------------
                    Name:   Stephen G. Locke
                    Title:  Assistant Vice President







<PAGE>

                                                                              11


                                                               SIGNATURE PAGE to
                                            THIRD AMENDMENT, CONSENT AND WAIVER,
                                                       dated as of July 19, 2001
                                                        to ALAMOSA HOLDINGS, LLC
                                           AMENDED AND RESTATED CREDIT AGREEMENT




                     To approve the Third Amendment, Consent and Waiver:


                     Name of Institution: SOCIETE GENERALE
                                          ------------------------

                                       By /s/ John Sadik-Khan
                                          ------------------------
                                          Name:   John Sadik-Khan
                                          Title:  Managing Director




                     To approve the extension of the above-named institution's
                     Term Commitment pursuant to Section 4(a) of the Amendment:


                     By /s/ John Sadik-Khan
                        -------------------------
                        Name:   John Sadik-Khan
                        Title:  Managing Director






<PAGE>

                                                                              12

                                                               SIGNATURE PAGE to
                                            THIRD AMENDMENT, CONSENT AND WAIVER,
                                                       dated as of July 19, 2001
                                                        to ALAMOSA HOLDINGS, LLC
                                           AMENDED AND RESTATED CREDIT AGREEMENT




              To approve the Third Amendment, Consent and Waiver:


              Name of Institution: IBM CREDIT CORP.
                                  ---------------------

                                By /s/ Thomas S. Curcio
                                   -----------------------
                                   Name: Thomas S. Curcio
                                   Title: Manager of Credit, Commercial and
                                          Specialty Financing




              To approve the extension of the above-named institution's Term
              Commitment pursuant to Section 4(a) of the Amendment:


              By /s/ Thomas S. Curcio
                 -----------------------
                 Name: Thomas S. Curcio
                 Title: Manager of Credit, Commercial and
                        Specialty Financing







<PAGE>

                                                                              13


                                                            SIGNATURE PAGE to
                                         THIRD AMENDMENT, CONSENT AND WAIVER,
                                                    dated as of July 19, 2001
                                                     to ALAMOSA HOLDINGS, LLC
                                        AMENDED AND RESTATED CREDIT AGREEMENT



                   To approve the Third Amendment, Consent and Waiver:


                   Name of Institution: FORTIS CAPITAL CORP.
                                       ---------------------------

                                     By  /s/ Colm Kelly
                                        --------------------------
                                        Name: Colm Kelly
                                        Title: Assistant Vice President




                   To approve the extension of the above-named institution's
                   Term Commitment pursuant to Section 4(a) of the Amendment:


                   By  /s/ Colm Kelly
                      -----------------------
                      Name:   Colm Kelly
                      Title:  Assistant Vice President







<PAGE>

                                                                              14

                                                            SIGNATURE PAGE to
                                         THIRD AMENDMENT, CONSENT AND WAIVER,
                                                    dated as of July 19, 2001
                                                     to ALAMOSA HOLDINGS, LLC
                                        AMENDED AND RESTATED CREDIT AGREEMENT




          To approve the Third Amendment, Consent and Waiver:


          Name of Institution: GENERAL ELECTRIC CAPITAL CORPORATION
                               ---------------------------------------

                                     By   /s/ Brian P. Ward
                                        -------------------------
                                        Name:   Brian P. Ward
                                        Title:  Manager-Operations




          To approve the extension of the above-named institution's Term
          Commitment pursuant to Section 4(a) of the Amendment:


          By /s/ Brian P. Ward
             ----------------------
             Name:   Brian P. Ward
             Title:  Manager-Operations





<PAGE>


                                                                              15


                                                            SIGNATURE PAGE to
                                         THIRD AMENDMENT, CONSENT AND WAIVER,
                                                    dated as of July 19, 2001
                                                     to ALAMOSA HOLDINGS, LLC
                                        AMENDED AND RESTATED CREDIT AGREEMENT



                    To approve the Third Amendment, Consent and Waiver:


                    Name of Institution: EXPORT DEVELOPMENT CORPORATION
                                         --------------------------------


                                      By   /s/ Luisa Rebolledo
                                          ------------------------
                                          Name:   Luisa Rebolledo
                                          Title:  Loan Asset Manager

                                      By   /s/ Dan Kovacs
                                           -----------------------
                                           Name:   Dan Kovacs
                                           Title:  Loan Asset Manager




                    To approve the extension of the above-named institution's
                    Term Commitment pursuant to Section 4(a) of the Amendment:


                    By   /s/ Luisa Rebolledo
                         ------------------------------
                         Name:   Luisa Rebolledo
                         Title:  Loan Asset Manager

                    By   /s/ Dan Kovacs
                         ------------------------------
                         Name:   Dan Kovacs
                         Title:  Loan Asset Manager






<PAGE>

                                                                              16

                                                             SIGNATURE PAGE to
                                          THIRD AMENDMENT, CONSENT AND WAIVER,
                                                     dated as of July 19, 2001
                                                      to ALAMOSA HOLDINGS, LLC
                                         AMENDED AND RESTATED CREDIT AGREEMENT




                           To approve the Third Amendment, Consent and Waiver:


                           Name of Institution: FRANKLIN FLOATING RATE TRUST
                                                ----------------------------
                                             By
                                               ------------------
                                               Name:
                                               Title:




                           To approve the extension of the above-named
                           institution's Term Commitment pursuant to
                           Section 4(a) of the Amendment:


                           By
                              --------------------------
                              Name:
                              Title:





<PAGE>

                                                                              17


                                                              SIGNATURE PAGE to
                                            THIRD AMENDMENT, CONSENT AND WAIVER,
                                                       dated as of July 19, 2001
                                                        to ALAMOSA HOLDINGS, LLC
                                           AMENDED AND RESTATED CREDIT AGREEMENT




                     To approve the Third Amendment, Consent and Waiver:


                     Name of Institution: WESTDEUTSCHE LANDESBANK GIROZENTRALE
                                          --------------------------------------

                                       By /s/ Michael J. Wynne
                                          ----------------------
                                           Name:  Michael J. Wynne
                                           Title: Managing Director



                                       By /s/ Cyril Dervaloy
                                          ----------------------
                                           Name:  Cyril Dervaloy
                                           Title: Associate Director



                     To approve the extension of the above-named institution's
                     Term Commitment pursuant to Section 4(a) of the Amendment:


                     By /s/ Michael J. Wynne
                        ----------------------------
                        Name:  Michael J. Wynne
                        Title: Managing Director




                      By /s/ Cyril Dervaloy
                         ----------------------
                          Name:  Cyril Dervaloy
                          Title: Associate Director