We report on observations of GRB 080503, a short gamma-ray burst (GRB) with very bright extended emission (about 30 times the gamma-ray fluence of the initial spike) in conjunction with a thorough comparison to other short Swift events. In spite of the prompt-emission brightness, however, the optical counterpart is extraordinarily faint, never exceeding 25 mag in deep observations starting at ~1 hr after the Burst Alert Telescope (BAT) trigger. The optical brightness peaks at ~1 day and then falls sharply in a manner similar to the predictions of Li & Paczyński (1998) for supernova-like emission following compact binary mergers. However, a shallow spectral index and similar evolution in X-rays inferred from Chandra observations are more consistent with an afterglow interpretation. The extreme faintness of this probable afterglow relative to the bright gamma-ray emission argues for a very low density medium surrounding the burst (a "naked" GRB), consistent with the lack of a coincident host galaxy down to 28.5 mag in deep Hubble Space Telescope imaging. The late optical and X-ray peak could be explained by a slightly off-axis jet or by a refreshed shock. Our observations reinforce the notion that short GRBs generally occur outside regions of active star formation, but demonstrate that in some cases the luminosity of the extended prompt emission can greatly exceed that of the short spike, which may constrain theoretical interpretation of this class of events. This extended emission is not the onset of an afterglow, and its relative brightness is probably either a viewing-angle effect or intrinsic to the central engine itself. Because most previous BAT short bursts without observed extended emission are too faint for this signature to have been detectable even if it were present at typical level, conclusions based solely on the observed presence or absence of extended emission in the existing Swift sample are premature.