Wide-field surveys are discovering a growing number of rare transients whose physical origin is not yet well understood. Here, we present optical and UV data and analysis of iPTF16asu, a luminous, rapidly-evolving, high velocity, stripped-envelope supernova. With a rest-frame rise-time of just 4 days and a peak absolute magnitude of $M_{rm g}=-20.4$ mag, the light curve of iPTF16asu is faster and more luminous than previous rapid transients. The spectra of iPTF16asu show a featureless, blue continuum near peak that develops into a Type Ic-BL spectrum on the decline. We show that while the late-time light curve could plausibly be powered by $^{56}$Ni decay, the early emission requires a different energy source. Non-detections in the X-ray and radio strongly constrain any associated gamma-ray burst to be low-luminosity. We suggest that the early emission may have been powered by either a rapidly spinning-down magnetar, or by shock breakout in an extended envelope of a very energetic explosion. In either scenario a central engine is required, making iPTF16asu an intriguing transition object between superluminous supernovae, Type Ic-BL supernovae, and low-energy gamma-ray bursts.