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 intermediate Palomar Transient Factory (iPTF) 16asu, a luminous, rapidly evolving, high-velocity, stripped-envelope supernova (SN). With a rest-frame rise time of just four days and a peak absolute magnitude of mag, the light curve of iPTF 16asu is faster and more luminous than that of previous rapid transients. The spectra of iPTF 16asu show a featureless blue continuum near peak that develops into an SN Ic-BL spectrum on the decline. We show that while the late-time light curve could plausibly be powered by 56Ni decay, the early emission requires a different energy source. Nondetections in the X-ray and radio strongly constrain the energy coupled to relativistic ejecta to be at most comparable to the class of low-luminosity gamma-ray bursts (GRBs). 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 iPTF 16asu an intriguing transition object between superluminous SNe, SNe Ic-BL, and low-luminosity GRBs.