Spin-triplet superconductors potentially host topological excitations that are of interest for quantum information processing. We report the discovery of spin-triplet superconductivity in UTe₂, featuring a transition temperature of 1.6 kelvin and a very large and anisotropic upper critical field exceeding 40 teslas. This superconducting phase stability suggests that UTe₂ is related to ferromagnetic superconductors such as UGe₂, URhGe, and UCoGe. However, the lack of magnetic order and the observation of quantum critical scaling place UTe₂ at the paramagnetic end of this ferromagnetic superconductor series. A large intrinsic zero-temperature reservoir of ungapped fermions indicates a highly unconventional type of superconducting pairing.