Inclusion Criteria:  Patients must be aged≥18 years with relapsed or refractory B-cell non-Hodgkin Lymphoma or chronic lymphocytic leukemia. Absolute CD3 count ≥50 mm^3. MRI brain and lumbar puncture with cerebral spinal fluid (CSF) analysis by cytology and flow cytometry without evidence of central nervous system (CNS) involvement ONLY in patients with history of CNS involvement or clinical suspicion at the time of enrollment. Measurable disease must be documented within 4 weeks of the time of consent defined as nodal lesions >15 mm in the long axis or extranodal lesions >10 mm in long OR bone marrow involvement that is biopsy proven. Karnofsky performance score ≥70. Adequate hepatic function, defined as aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤3 x upper limit of normal (ULN); serum bilirubin and alkaline phosphatase ≤3 x ULN, or considered not clinically significant as per the clinical PI's discretion (e.g., Gilbert's or indirect hyperbilirubinemia) or felt to be due to underlying disease. ANC≥1000 with no G-CSF within 72 hours or pegylated G-CSF within 10 days Platelets≥50,000 with no transfusion within 72 hours of eligibility testing Adequate renal function, defined as creatinine clearance≥60 ml/min AND serum Cr≤1.5 mg/dL Able to provide written informed consent. Agree to practice birth control during the study. Adequate cardiac function as indicated by New York Heart Association (NYHA) classification I or II AND left ventricular ejection fraction of ≥45% (by cardiac echocardiogram or multigated acquisition scan) and adequate pulmonary function as indicated by room air oxygen saturation of ≥90%. Expected survival >12 weeks. Negative urine or serum pregnancy test in females of child bearing potential at study entry. Meet criteria for regarding fertility and contraception. No contraindication to central line access. Diagnosis of CLL or B-cell NHL including Follicular Lymphoma, Marginal Zone Lymphoma (splenic, nodal, extranodal), Mantle Cell Lymphoma, Burkitt Lymphoma, and DLBCL with associated subtypes (high-grade or aggressive B-cell lymphoma, T-cell/histocyte rich B-cell lymphoma, primary mediastinal B-cell lymphoma, Epstein-Barr virus (EBV)+ diffuse large B-cell lymphoma, transformed lymphoma such as transformed follicular or marginal zone, and Richter's transformation). For CLL patients, must have active, measurable disease (>5% CLL involvement of marrow or nodal disease as noted above) and failed/progressed or been intolerant to both Bruton Tyrosine Kinase (BTK) inhibitors (ibrutinib, acalabruinitib, or zanabrutinib) and BCL2 inhibitors (e.g. venetoclax).  For B-cell NHL Patients must have active, measurable disease as defined and meet one of the following criteria:  Must have received an anti-CD20 monoclonal antibody and at minimum two different chemotherapy regimens appropriate for their disease and be ineligible per treating physician to receive autologous transplant Relapse post-autologous transplant Relapse post-allogeneic transplant  Exclusion Criteria:  Positive beta-human chorionic gonadotropin (HCG) in female of child-bearing potential defined as per table 1. Confirmed active human immunodeficiency virus (HIV), Hepatitis B or C infection. History of significant autoimmune disease OR active, uncontrolled autoimmune phenomenon requiring steroid therapy defined as >20 mg of prednisone or equivalent daily. Presence of ≥grade 3 non-hematologic toxicities as per CTCAE version 5.0 from any previous treatment unless it is felt to be due to underlying disease. Concurrent use of investigational therapeutic agents or enrollment on another therapeutic clinical trial at any institution. Minimum of 14 days or 5 half-lives of the drug (whichever is shorter) washout prior to apheresis. Refusal to participate in the long-term follow-up protocol.  Patients with active CNS involvement by malignancy on MRI or by lumbar puncture.  a. Patients with prior CNS disease that has been effectively treated will be eligible providing treatment was >4 weeks before enrollment and a remission documented within 8 weeks of planned CAR T-cell infusion by MRI brain and CSF analysis.  Previous recipients of allogeneic hematopoietic stem cell transplantation (AHCT) are excluded if they are <100 days' post-transplant, have evidence of active graft-versus-host-disease (GVHD) of any grade, or are currently on immunosuppression. Anti-CD20 directed treatment within 4 weeks of cell infusion. Anti-CD19 directed treatment within 4 weeks of cell infusion. Anti-CD22 directed treatment within 4 weeks of cell infusion Cytotoxic chemotherapy other than lymphodepletion within 14 days of CAR T-cell infusion  Targeted oral agents or antibody directed treatment within 7 days of apheresis  a. Other than BTK inhibitors which can be continued until day prior to apheresis  Cytotoxic chemotherapy treatment within 10 days or steroid treatment (other than replacement dose steroids) within 7 days prior to apheresis collection for CAR T-cells.  Oral chemotherapeutic agents or antibody directed treatment within 7 days of apheresis  a. BTK inhibitors are allowed until 1-day prior to apheresis and can re-start until 1-day prior to lymphodepletion  Patients post solid organ transplant who develop high grade lymphomas or leukemias. Concurrent active malignancy other than basal or squamous cell carcinomas of the skin. Prior anti-CD19, anti-CD20, or anti-CD22 CAR T-cell therapy (autologous or allogeneic).  Special Criteria for regarding Fertility and Contraception:  Female subjects of reproductive potential (women who have reached menarche or women who have not been post-menopausal for at least 24 consecutive months, i.e., who have had menses within the preceding 24 months, or have not undergone a sterilization procedure [hysterectomy or bilateral oophorectomy]) must have a negative serum or urine pregnancy test performed as part of eligibility criteria. Lactating women are eligible for this study but will be asked to not provide breast milk to their child from Day -4 through Day +90 after CAR T-cell therapy. It is possible they may no longer be able to lactate after receiving chemotherapy and treatment.  Due to the high-risk level of this study, while enrolled, all subjects must agree not to participate in a conception process (e.g., active attempt to become pregnant or to impregnate, sperm donation, in vitro fertilization). Additionally, if participating in sexual activity that could lead to pregnancy, the study subject must agree to use reliable and double barrier methods of contraception during the follow-up period of the protocol.  Acceptable birth control includes a combination of two of the following methods:  Condoms (male or female) with or without a spermicidal agent. Diaphragm or cervical cap with spermicide. Intrauterine device (IUD). Hormonal-based contraception.  Subjects who are not of reproductive potential (women who are premenarche or have been post-menopausal for at least 24 consecutive months or have undergone hysterectomy tubal ligation, salpingectomy, and/or bilateral oophorectomy or men who have documented azoospermia) are eligible without requiring the use of contraception.