  We set the strongest limits to-date on the velocity-independent dark matter
(DM) - proton cross section $\sigma$ for DM masses $m = 10\,\mathrm{keV}$ to
$100\,\mathrm{GeV}$, using large-scale structure traced by the Lyman-alpha
forest: e.g., a 95% lower limit $\sigma < 6 \times 10^{-30}\,\mathrm{cm}^2$,
for $m = 100\,\mathrm{keV}$. Our results complement direct detection, which has
limited sensitivity to sub-GeV DM. We use an emulator of cosmological
simulations, combined with data from the smallest cosmological scales used
to-date, to model and search for the imprint of primordial DM-proton
collisions. Cosmological bounds are improved by up to a factor of 25.
