Intraperitoneal FT536 in Recurrent Ovarian, Fallopian Tube, and Primary Peritoneal Cancer

Inclusion Criteria: * Recurrent epithelial ovarian cancer, fallopian tube, or primary peritoneal cancer meeting one of the following minimal prior treatment requirements (no limit to the maximum number of prior treatments). * Must have received prior bevacizumab. * In the presence of a BRCA mutation, must have received a prior PARP inhibitor. * Adequate organ function within 14 days (28 days for pulmonary and cardiac) of study treatment (CY/Flu) start * Agrees to the placement of an intraperitoneal catheter before the 1st dose of study directed drug (chemotherapy) and remains in place through Day 36 or longer if retreatment is planned. Refer to Section 6.4 if catheter cannot be successfully placed. * Agrees to undergo a tumor biopsy if feasible at the time the catheter is placed and removed - Accessible tumor for biopsy is not required for eligibility * Must agree to and sign the consent for the companion Long-Term Follow-Up study (CPRC# 2021LS077) to fulfill the FDA recommended 15 years of followup for a genetically modified cell product. Exclusion Criteria: * Pregnant or breastfeeding or planning on becoming pregnant in the next 6 months. If of childbearing potential (have a uterus and ovaries) and engaged in heterosexual intercourse, must have a negative pregnancy test (serum or urine) within 14 days before the 1st CY/Flu. Patient must agree to use highly effective method of birth control from the screening visit until at least 12 months after the final dose of CY, or at least 4 months after the final dose of FT536, whichever is longer. * Currently receiving or likely to require systemic immunosuppressive therapy (e.g., prednisone \>5 mg daily) for any reason from Day -5 to 14 days after the last FT536 infusion) with the exception of corticosteroids as a pre-medication per institutional standard of care - topical and inhaled steroids are permitted. * Active autoimmune disease requiring systemic immunosuppressive therapy. * History of severe asthma and currently on chronic systemic medications. * Uncontrolled bacterial, fungal or viral infections with progression of clinical symptoms despite therapy. * Receipt of any biological therapy, chemotherapy, or radiation therapy (except palliative RT), within 2 weeks prior to the first dose of FT536 or five half-lives, whichever is shorter; or any investigational agent within 28 days prior to the first dose of FT536. * Live vaccine within 6 weeks prior to start of lympho-conditioning. * Known allergy to the following FT536 components: albumin (human) or dimethyl sulfoxide (DMSO). * Prior enoblituzumab. * Non-malignant CNS disease such as stroke, epilepsy, CNS vasculitis, or neurodegenerative disease or receipt of medications for these conditions in the 2-year period leading up to study enrollment. (Refer to Section 5.1.8 regarding history of brain metastases.) * Known history of HIV positivity or active hepatitis C or B - chronic asymptomatic viral hepatitis is allowed. * Presence of any medical or social issues that are likely to interfere with study conduct or may cause increased risk to patient. * Any medical condition or clinical laboratory abnormality that, per investigator judgement, precludes safe participation in and completion of the study or that could affect compliance with protocol conduct or interpretation of results.