Every library has a different method of counting these things. My customer wants to keep the book longer. Almost all ILL books can be renewed , once. ILL will ask the loaning library, usually electronically, for the renewal. ILL will let you know the new due date, or if the book must be returned.
Some libraries have a non-renewal policy .
Usually ILL will put this as a note on the form notifying you that the item will be coming from this library. Syracuse University never allows renewals. Videocassettes usually cannot be renewed . You are a library, therefore you may get an interlibrary loan request directly. It's up to you whether to loan or not.