  The center of the Milky Way is a prime site to search for signals of dark
matter (DM) annihilation due to its proximity and expected high concentration
of DM. The amplification of the dispersion velocity of DM particles in the
Galactic center (GC), caused by baryonic contraction and feedback, makes this
particular region of the sky an even more promising target for exploring
velocity-dependent DM models. Here we demonstrate that current GC observations
with the H.E.S.S. telescope, presently the most sensitive TeV-scale gamma-ray
telescope in operation in this region of the sky, set the strongest constraints
on velocity-dependent annihilating DM particles with masses above 200 GeV. For
p-wave annihilations, they improve the current constraints by a factor of
$\sim$4 for a DM mass of 1 TeV. For the spatial distribution of DM, we use the
results of the latest FIRE-2 zoom cosmological simulation of Milky Way-size
halos. In addition, we utilize the newest version of the GALPROP cosmic-ray
propagation framework to simulate the Galactic diffuse gamma-ray emission in
the GC. We have found that p-wave (d-wave) DM particles with a mass of
approximately 1.7 TeV and annihilating into the $W^+$$W^-$ channel exhibit a
velocity-weighted annihilation cross-section upper limit of 4.6$\times$
10$^{-22}$ cm$^3$s$^{-1}$ (9.2$\times$10$^{-17}$ cm$^3$s$^{- 1}$) at a 95\%
confidence level. This is about 460 (2$\times$ 10$^{6}$) times greater than the
thermal relic cross-section for p-wave (d-wave) DM models.
