  We present the Pan-STARRS1 discovery of PS1-10afx, a unique
hydrogen-deficient superluminous supernova (SLSN) at z=1.388. The light curve
peaked at z_P1=21.7 mag, making PS1-10afx comparable to the most luminous known
SNe, with M_u = -22.3 mag. Our extensive optical and NIR observations indicate
that the bolometric light curve of PS1-10afx rose on the unusually fast
timescale of ~12 d to the extraordinary peak luminosity of 4.1e44 erg/s (M_bol
= -22.8 mag) and subsequently faded rapidly. Equally important, the SED is
unusually red for a SLSN, with a color temperature of 6800 K near maximum
light, in contrast to previous H-poor SLSNe, which are bright in the UV. The
spectra more closely resemble those of a normal SN Ic than any known SLSN, with
a photospheric velocity of 11,000 km/s and evidence for line blanketing in the
rest-frame UV. Despite the fast rise, these parameters imply a very large
emitting radius (>5e15 cm). We demonstrate that no existing theoretical model
can satisfactorily explain this combination of properties: (i) a nickel-powered
light curve cannot match the combination of high peak luminosity with the fast
timescale; (ii) models powered by the spindown energy of a rapidly-rotating
magnetar predict significantly hotter and faster ejecta; and (iii) models
invoking shock breakout through a dense circumstellar medium cannot explain the
observed spectra or color evolution. The host galaxy is well detected in
pre-explosion imaging with a luminosity near L*, a star formation rate of 15
M_sun/yr, and is fairly massive (2e10 M_sun), with a stellar population age of
1e8 yr, also in contrast to the dwarf hosts of known H-poor SLSNe. PS1-10afx is
distinct from known examples of SLSNe in its spectra, colors, light-curve
shape, and host galaxy properties, suggesting that it resulted from a different
channel than other hydrogen-poor SLSNe.
