A Phase I Trial of Anti-CC Chemokine Receptor 4 Chimeric Antigen Receptor T Cells (CCR4 CAR T Cells) for CCR4 Expressing T-cell Malignancies Including Peripheral T-cell Non-Hodgkin Lymphoma (PTCL) and Cutaneous T-cell Non-Hodgkin Lymphoma (CTCL)

* INCLUSION CRITERIA: * Pathologically (biopsy) confirmed histologic diagnosis of a relapsed/refractory CCR4+ mature T-cell malignancy from one of the following subtypes: peripheral T-cell lymphoma not otherwise specified (PTCL-NOS), angioimmunoblastic T-cell lymphoma (AITL), anaplastic large cell lymphoma (ALCL), hepatosplenic t-cell lymphoma (HSTCL), monomorphic epithelialtropic intestinal lymphoma (MEITL), enteropathy associated T-cell lymphoma (EATL) or cutaneous T-cell lymphoma (CTCL) including mycosis fungoides and subacute panniculitis-like T-cell Lymphoma, or lymphomatous subtypes of ATL without evidence of CNS involvement or substantial circulating disease confirmed by the Laboratory of Pathology, NCI. --CCR4+ is defined as \>= 10% malignant cells positive for CCR4 by immunohistochemistry. It is preferred to have a fresh biopsy to confirm the CCR4 status. In the event a fresh biopsy cannot be safely performed in the opinion of the treating physician, an archival biopsy sample taken at the time of previous progression can be used. * Adequate tissue \[a formalin fixed tissue block or 15 slides of tumor sample (archival or fresh)\] from diagnostic biopsy (archival or fresh) must be available. NOTE: Tissue will be used for assessment of CCR4 expression on malignant cells by immunohistochemistry with any leftover slides or samples to be used for correlative studies. Tumor tissue may be from any previously collected tissue and adequacy is at the discretion of the Principal Investigator. If prior tissue is not available, a screening biopsy will be necessary unless repeat biopsy is deemed unsafe by the treating physician in consultation with the Principal Investigator. * Participants must have disease that is relapsed or refractory after prior therapy as follows: * Participants with ALCL must have failed at least one prior line of Brentuximab-containing therapy. * Due to the generally indolent nature of the disease, participants with Mycosis Fungoides must have exhausted all standard therapies as determined by the enrolling physician and principal investigator to be eligible for this study. * All other participants must have failed at least two lines of prior therapy. * Participants must have measurable or evaluable disease at the time of enrollment. For participants with systemic T-cell lymphoma, this is defined by any evidence from CT scan or PET-CT-avid disease based on the Lugano criteria. For participants with Cutaneous T-cell Lymphoma, positive scores based on Modified Severity-Weighted Assessment Tool (mSWAT) criteria are acceptable. * Participants must be \>=18 years of age at the time of signing informed consent. * Adequate performance status (PS) as follows: ECOG PS 0-1. * Adequate organ function as evidenced by the following laboratory parameters: * Absolute neutrophil count (ANC) \>= 1,000 /microL * Platelets \>= 75,000 / microL * Hemoglobin (Hgb) \>= 9 g/dL (transfusions permitted) * Creatinine Clearance \>= 60 mL/min/1.73m\^2 per Cockcroft Gault equation; For participants \< 60 per Cockcroft Gault a direct measurement may be used * Serum total bilirubin \<= 3 X upper limit of normal (ULN) * Aspartate Aminotransferase (AST) and Alanine Aminotransferase (ALT) \<= 3 X ULN * Left ventricular ejection fraction \> 50% by echocardiogram performed * ECG No clinically significant ECG findings (Arrhythmias or evidence of ischemic heart disease with clinical correlate) Note: Participants with well-controlled atrial fibrillation are eligible. --FEV1 and DLCO \> 60% of predicted (adjustment for Hgb acceptable) -Individuals of child-bearing potential (IOCBP) must have a negative urine or blood HCG pregnancy test at screening. NOTE: IOCBP is defined as any person assigned female at birth who has experienced menarche and who has not undergone successful surgical sterilization or who is not postmenopausal. -Individuals of child-bearing potential (IOCBP) must agree to use highly effective contraception (hormonal, intrauterine device \[IUD\], abstinence, surgical sterilization) or practice abstinence starting at the time of study entry, for the duration of study therapy, and 12 months after the last dose of combined chemotherapy. Individuals able to father a child must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) for the duration of the study treatment and for 4 months after the last dose of combined chemotherapy. We also will recommend these individuals with partners of childbearing potential ask partners to be on highly effective birth control (hormonal, IUD, surgical sterilization) * Nursing participants must be willing to discontinue nursing through 12 weeks after cell infusion. * Potential participants must agree to stay within 1-hour drive of NIH clinical center from date of initial discharge from hospitalization (no earlier than D+15) through initial D+28 follow-up and be willing and able to return for in-person follow-up visits through month 3 of the study. * Ability of participant or Legally Authorized Representative (LAR) to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: * Participants with any current or prior CNS involvement by malignancy are excluded from this study. All potential participants will be screened with brain imaging prior to enrollment on study. * Participants with \>1000 atypical cells/mm\^3 by peripheral blood flow cytometry at screening. * Participants with a history of serologically or biopsy confirmed autoimmune disorders are excluded from this study. As an exception, participants with EATL whose celiac disease is well controlled and who will maintain a strict gluten-free diet are eligible. Participants with prior autoimmune thyroiditis who are now on stable thyroid replacement therapy are also eligible. * HTLV I/II positive participants with a history of HTLV-associated myelopathy/tropical spastic paraparesis (TSP) * Participants who have received prior CD25-directed therapy. * Current or prior anti-cancer treatment prior to the first dose of study drug as defined below: * Any cytotoxic therapy, immunotherapy, antitumor vaccines or monoclonal antibodies within 2 weeks before the start of lymphodepleting chemotherapy. * High doses of systemic corticosteroids (\>20 mg prednisone or equivalent) 5 days before apheresis and/or 5 days before CAR T cell infusion. * Participants who have not reached D+100 following auto-SCT or who have any unresolved Auto-SCT related complications (e.g. pneumonitis). * Participants who have undergone prior allogeneic stem cell at any time. * Participants taking any investigational agents for any disease/ condition. * Seropositive for human immunodeficiency virus (HIV). * Active bacterial infections or active viral infections (CMV, syphilis) * Uncontrolled EBV infection Note: EBV positive test is allowed due to frequent association of active EBV with mature T-cell malignancies, which frequently resolve with improved control of the malignancy. EBV positive participants may be treated with rituximab or biosimilar prior to lymphodepleting chemotherapy at investigator s discretion. * Active hepatitis C infection. NOTE: Participants seropositive for hepatitis C virus (HCV) infection must have been treated and cured as defined by undetectable HCV viral load. -Active hepatitis B infection. NOTE: Participants that are positive for hepatitis B core antibody (HBcAb) or hepatitis B surface antigen (HBsAg) must have a negative hepatitis B virus polymerase chain reaction (HBV PCR) result \<100 IU/mL at screening. Those who are HBV PCR positive are excluded. Those hepatitis B surface antigen (HBsAg) or hepatitis B core antibody (HBcAb) positive with a negative PCR for hepatitis B will be treated with antivirals designed to prevent hepatitis B reactivation (e.g., entecavir) and have monitoring for hepatitis B reactivation with PCR. * Participants with current cardiac atrial or cardiac ventricular lymphoma involvement. * History of myocardial infarction, cardiac angioplasty or stenting, unstable angina, New York Heart Association Class II or greater congestive heart failure, or other clinically significant cardiac disease within 12 months of enrollment. * History of idiopathic pulmonary fibrosis, organizing pneumonia (e.g. bronchiolitis obliterans), drug-induced pneumonitis, idiopathic pneumonitis, or evidence of active pneumonitis per chest computer tomography (CT) scan at screening. NOTE: History of radiation pneumonitis in the radiation field (fibrosis) is allowed. * History or presence of non-malignant CNS disorder such as seizure disorder, cerebrovascular ischemia/hemorrhage, dementia, cerebellar disease, or any autoimmune disease with CNS involvement * Deep vein thrombosis or pulmonary embolism requiring ongoing systemic anticoagulation * History of severe immediate hypersensitivity reaction to tocilizumab or any of the agents used in this study * Participants with second malignancies in addition to their T-cell malignancy are not eligible if the second malignancy has required treatment (including maintenance therapy) within the past 3 years or is not in complete remission. There are two exceptions to this criterion: successfully treated non-metastatic basal cell or squamous cell skin carcinoma. * Uncontrolled intercurrent illness including, but not limited to the following that may limit interpretation of results or that could increase risk to the participant.