Umbilical & Cord Blood (CB) Derived CAR-Engineered NK Cells for B Lymphoid Malignancies

If you are reading and signing this form on behalf of a potential participant, please note: Any time the words "you," "your," "I," or "me" appear, it is meant to apply to the potential participant. The goal of this clinical research study is to learn if giving genetically changed immune cells, called CAR-NK cells, after chemotherapy will improve the disease in stem cell transplant patients with relapsed (has returned) and/or refractory (has not responded to treatment) B-cell lymphoma or leukemia. Also, researchers want to find the highest tolerable dose of CAR-NK cells to give to patients with relapsed or refractory B-cell lymphoma or leukemia. The safety of this treatment will also be studied. This is an investigational study. The making of and infusion of genetically changed NK cells and the drug AP1903 (if you receive it, explained below) are not FDA approved or commercially available for use in this type of disease. They are currently being used for research purposes only. The chemotherapy drugs in this study (fludarabine, cyclophosphamide, and mesna) are commercially available and FDA approved. Up to 36 patients will take part in this study. All will be enrolled at MD Anderson.