  Huge efforts have been deployed to detect dark matter (DM) in the GeV-TeV
mass range involving various detection techniques, and led to strong
constraints in the available parameter space. We compute here the sensitivity
to sub-GeV DM that can be probed from the inevitable cosmic-ray scattering onto
DM particles populating the Milky Way halo. Inelastic scattering of energetic
cosmic rays off DM would produce high-energy gamma rays in the final state,
providing a new avenue to probe the poorly-constrained so far sub-GeV dark
matter mass range. In this work we derive sensitivity forecasts for the
inelastic cosmic-ray proton - DM cross section for current and future
very-high-energy gamma-ray observatories such as H.E.S.S., LHAASO, CTA and SWGO
in the 100 eV to 100 MeV mass range. These inelastic cross section constraints
are converted to the elastic proton - DM cross section to highlight further
complementarity with cosmological, collider and direct detection searches. The
sensitivity computed at 95\% confidence level on the elastic cross section
reaches $\sim$2$\times$ 10$^{-32}$ cm$^2$ for a 100 keV DM mass for
H.E.S.S.-like and $\sim$7$\times$ 10$^{-34}$ cm$^2$ for a $\sim$1 keV DM mass
for LHAASO. The sensitivity prospects for CTA and a strawman SWGO model reach
$\sim$6$\times$ 10$^{-34}$ cm$^2$ and $\sim$4$\times$ 10$^{-35}$ cm$^2$, for DM
masses of 10 keV and 1 keV, respectively. The sensitivity reach of the
gamma-ray observatories considered here enables to probe an uncharted region of
the DM mass - cross section parameter space.
