Inclusion Criteria:  Chest radiograph or computed tomography (CT) of the thorax approximately 12 months before screening not demonstrating any abnormality considered to be significantly contributing to the chronic cough Have a diagnosis of refractory chronic cough (RCC) or unexplained chronic cough (UCC) for at least one year Women of childbearing potential and their male partners must use 2 acceptable methods of contraception Male subjects and their female partners of childbearing potential must use 2 acceptable methods of contraception Have provided written informed consent  Exclusion Criteria:  Positive diagnostic nucleic acid test for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) Current smoker (including e-cigarettes), individuals who have given up smoking within the past 12 months, or individuals with a smoking history of 20 pack-years Treatment with an ACE-inhibitor as the potential cause of a subject's cough or requiring treatment with an ACE-inhibitor during the study or within 12 weeks prior to the Baseline Visit History of upper or lower respiratory tract infection or recent significant change in pulmonary status within 4 weeks History of cystic fibrosis Positive test for any drug of abuse History of malignancy within 5 years prior to the Baseline Visit History of infection or known active infection with human immunodeficiency (HIV), hepatitis B virus (HBV), or hepatitis C virus (HCV) History of hypersensitivity or intolerance to AX-8 or other TRPM8 agonists or any of the excipients