CAR.CD19-CD28-zeta-2A-iCasp9-IL15-Transduced Cord Blood NK Cells, High-Dose Chemotherapy, and Stem Cell Transplant in Treating Participants With B-cell Lymphoma

PRIMARY OBJECTIVES: I. To establish the safety and relative efficacy of CAR.CD19-CD28-zeta-2A-iCasp9-IL15-transduced cord blood natural killer (CB-NK) cells in patients with B cell non-Hodgkin lymphoma (NHL) undergoing high dose chemotherapy and autologous stem cell transplantation. SECONDARY OBJECTIVES: I. To estimate the relapse-free survival (RFS). II. To estimate the overall survival (OS). III. To quantify the persistence of infused CAR.CD19-CD28-zeta-2A-iCasp9-IL15-transduced CB-NK cells in the recipient. OUTLINE: This is a phase I, dose-escalation study of CAR.CD19-CD28-zeta-2A-iCasp9-IL15-transduced CB-NK cells followed by a phase II study. Participants receive rituximab intravenously (IV) over 3 hours on days -14 and -8, carmustine IV over 2 hours on day -13, etoposide IV over 3 hours twice daily (BID) on days -12 to -9, cytarabine IV over 1 hour BID on days -12 to -9, melphalan IV over 30 minutes on day -8, CAR.CD19-CD28-zeta-2A-iCasp9-IL15-transduced CB-NK cells IV over 1 hour on day -5. Participants undergo autologous stem cell transplantation (ASCT) on day 0. Beginning day 0, participants receive filgrastim subcutaneously (SC) once daily (QD) until evidence of an absolute neutrophil count (ANC) of 0.5 x 10\^9/L per 3 consecutive days. After completion of study treatment, participants are followed for up to 15 years.