We use the latest measurements of the Milky Way satellite population from the Dark Energy Survey and Pan-STARRS1 to infer the most stringent astrophysical bound to date on velocity-dependent interactions between dark matter particles and protons. We model the momentum-transfer cross section as a power law of the relative particle velocity $v$ with a free normalizing amplitude, $sigma_text{MT}=sigma_0 v^n$, to broadly capture the interactions arising within the non-relativistic effective theory of dark matter-proton scattering. The scattering leads to a momentum and heat transfer between the baryon and dark matter fluids in the early Universe, ultimately erasing structure on small physical scales and reducing the abundance of low-mass halos that host dwarf galaxies today. From the consistency of observations with the cold collisionless dark matter paradigm, using a new method that relies on the most robust predictions of the linear perturbation theory, we infer an upper limit on $sigma_0$ of $1.4times10^{-23}$, $2.1times10^{-19}$, and $1.0times10^{-12} mathrm{cm}^2$, for interaction models with $n=2,4,6$, respectively, for a dark matter particle mass of $10 mathrm{MeV}$. These results improve observational limits on dark matter--proton scattering by orders of magnitude and thus provide an important guide for viable sub-GeV dark matter candidates.
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