Engineered immune cells take on rare leukemia

Copied word for word from the study's registry entry, so the wording is the study team's rather than ours.

* INCLUSION CRITERIA * Histologically confirmed diagnosis of HCL or HCLv according to morphological and immunophenotypic criteria of WHO classification \[WHO, 2008 revised 2016\] of lymphoid neoplasm. * Participants should have any of the following indications for therapy: * ANC \<1/nL, * Hemoglobin \<10g/dL, * Platelets\<100/nL, * Symptomatic splenomegaly, * HCL mass with short axis \> 2 cm outside or \>0.5 cm inside the CNS, * HCL/HCLv count \>5/nL in blood or \>25/mm\^3 in CSF, * HCL/HCLv count doubling time \<6 months and increasing lytic or blastic bone lesions Participants who have eligible blood counts within 4 weeks from the initiation of study will not be considered ineligible if subsequent blood counts prior to enrollment fluctuate and become ineligible up until the time of enrollment. * HCL/HCLv, after prior treatment with, ineligible for, refusal of, or inability to obtain 1)rituximab given concurrently with or sequentially after purine analog, 2) moxetumomab pasudotox-tdft, and 3) BRAF-inhibition. * CD22 expression must be detected on greater than 80% of malignant cells by flow cytometry. * Participants must have measurable or evaluable disease at the time of enrollment, which may include any evidence of disease including minimal residual disease (MRD) detected by flow cytometry or immunohistochemistry. * Age \>=18 years * ECOG performance \<=2 (Karnofsky \>=60%, see Appendix A), participants are exempt from this criterion if poor performance status is related to HCL. * Participants must have adequate organ function as defined below: Participants must have recovered from the acute side effects of their prior therapy, such that eligibility criteria are met. If participants exhibit minor lab abnormalities that are determined to be related to HCL (not therapy-related), then those participants will be allowed to participate * Total bilirubin \<= 3 ULN, unless consistent with Gilbert s (ratio between total and direct bilirubin \> 5) * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<= 3x upper limit of normal (ULN) * Alkaline phosphatase \< 2.5 ULN * Serum creatinine \<= 1.5 mg/dL or creatinine clearance \>= 60 mL/min/1.73 m\^2 for participants with creatinine levels above institutional normal calculated using eGFR or measured * Serum albumin \> 2 g/dL * Prothrombin time (PT)/International Normalized Ratio (INR) \< 2.5x ULN (if on warfarin, PT/INR \< 3.5x ULN; If on any other anticoagulation, PT \< 2.5x ULN * Fibrinogen \>= 0.5x lower limit of normal * Participants with CNS disease are eligible, with exceptions * Participants with history of allogeneic stem cell transplantation are eligible if at least 100 days post-transplant, if there is no evidence of active graft-versus-host disease (GVHD) and no longer taking immunosuppressive agents for at least 30 days prior to initiation of study intervention. * Women of childbearing potential (WOCBP) must agree to use effective contraception (barrier, hormonal, intrauterine device \[IUD\], abstinence, surgical sterilization) at the study entry and up to 12 months after the last dose of combined chemotherapy or 4 months after cells infusion, whichever is later. Men must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) at the study entry and up to 4 months after the last dose of study drug. * Breastfeeding participants must be willing to discontinue breastfeeding from study treatment initiation through 4 months after the last dose of study drug. * Ability of participant to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA * Pregnancy * Systemic chemotherapy, immunotherapy, or radiation therapy \<= 2 weeks prior to apheresis with the following exception: * Participants receiving steroids may be enrolled, provided there has been no increase in dose for at least 1 week prior to starting apheresis; * For radiation therapy: Radiation therapy must have been completed at least 3 weeks prior to enrollment (including CNS radiation), with the exception that there is no time restriction if the volume of bone marrow treated is less than 10% and also the participant has measurable/evaluable disease outside the radiation port. * Other anti-neoplastic investigational agents, or antibody-based therapies currently or within 2 weeks prior to apheresis * Participants taking warfarin * Prior CAR therapy within 30 days prior to apheresis or prior CAR therapy at any time with evidence for persistence of CAR T cells in blood samples (circulating levels of genetically modified cells of \>= 5% by flow cytometry) * Seropositive for human immunodeficiency virus (HIV) antibody. (Participants with HIV are at increased risk of lethal infections when treated with marrow-suppressive therapy. Appropriate studies will be undertaken in participants receiving combination antiretroviral therapy in the future should study results indicate effectiveness.) * Seropositive for hepatitis C virus (HCV) or positive for hepatitis B surface antigen (HbsAG). Participants who convert to negative will not be excluded for history of positive test. * Uncontrolled, symptomatic, intercurrent illness including but not limited to infection, congestive heart failure, unstable angina pectoris, cardiac arrhythmia, asthma, chronic obstructive pulmonary disease, psychiatric illness, or social situations that would limit compliance with study requirements or in the opinion of the PI would pose an unacceptable risk to the subject * History of severe, immediate hypersensitivity reaction attributed to compounds of similar chemical or biologic composition to any agents used in study or in the manufacturing of the cells (i.e., gentamicin)