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$^*$.
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