  We report on observations of GRB 080503, a short gamma-ray burst 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 BAT trigger. The optical brightness peaks at ~1 day and then
falls sharply in a manner similar to the predictions of Li & Paczynski (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
HST imaging. 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. 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. (abridged)
