The formation of relativistic jets by an accreting compact object is one of the fundamental mysteries of astrophysics. Although the theory is poorly understood, observations of relativistic jets from systems known as microquasars (compact binary stars)(1,2) have led to a well established phenomenology(3,4). Relativistic jets are not expected to be produced by sources with soft or supersoft X-ray spectra, although two such systems are known to produce relatively low-velocity bipolar outflows(5,6). Here we report the optical spectra of an ultraluminous supersoft X-ray source (ULS7,8) in the nearby galaxy M81 (M81 ULS-1; refs 9, 10). Unexpectedly, the spectra show blueshifted, broad H alpha emission lines, characteristic of baryonic jets with relativistic speeds. These time-variable emission lines have projected velocities of about 17 per cent of the speed of light, and seem to be similar to those from the prototype microquasar SS 433 (refs 11, 12). Such relativistic jets are not expected to be launched from white dwarfs(13), and an origin from a black hole or a neutron star is hard to reconcile with the persistence of M81 ULS-1's soft X-rays(10). Thus the unexpected presence of relativistic jets in a ULS challenges canonical theories of jet formation(3,4), but might be explained by a long-speculated, supercritically accreting black hole with optically thick outflows(14-20).