  Understanding exotic, non s--wave--like states of Cooper pairs is important
and may lead to new superconductors with higher critical temperatures and novel
properties. Their existence is known to be possible but has always been thought
to be associated with non--traditional mechanisms of superconductivity where
electronic correlations play an important role. Here we use a first principles
linear response calculation to show that in doped Bi$_{2}$Se$_{3}$ an
unconventional p--wave--like state can be favored via a conventional
phonon--mediated mechanism, as driven by an unusual, almost singular behavior
of the electron--phonon interaction at long wavelengths. This may provide a new
platform for our understanding superconductivity phenomena in doped band
insulators.
