  Indirect searches for dark matter through Standard Model products of its
annihilation generally assume a cross-section which is dominated by a term
independent of velocity ($s$-wave annihilation). However, in many DM models an
$s$-wave annihilation cross-section is absent or helicity suppressed. To
reproduce the correct DM relic density in these models, the leading term in the
cross section is proportional to the DM velocity squared ($p$-wave
annihilation). Indirect detection of such $p$-wave DM is difficult because the
average velocities of DM in galaxies today are orders of magnitude slower than
the DM velocity at the time of decoupling from the primordial thermal plasma,
suppressing the annihilation cross-section today by some five orders of
magnitude relative to its value at freeze out. Thus $p$-wave DM is out of reach
of traditional searches for DM annihilations in the Galactic halo. Near the
region of influence of a central supermassive black hole, such as Sgr A$^*$,
however, DM can form a localized over-density known as a `spike'. In such
spikes the DM is predicted to be both concentrated in space and accelerated to
higher velocities, allowing the $\gamma$-ray signature from its annihilation to
potentially be detectable above the background. We use the $Fermi$ Large Area
Telescope to search for the $\gamma$-ray signature of $p$-wave annihilating DM
from a spike around Sgr A$^*$ in the energy range 10 GeV-600 GeV. Such a signal
would appear as a point source and would have a sharp line or box-like spectral
features difficult to mimic with standard astrophysical processes, indicating a
DM origin. We find no significant excess of $\gamma$ rays in this range, and we
place upper limits on the flux in $\gamma$-ray boxes originating from the
Galactic Center. This result, the first of its kind, is interpreted in the
context of different models of the DM density near Sgr A$^*$.
