  Starburst galaxies are well-motivated astrophysical emitters of high-energy
gamma-rays. They are well-known cosmic-ray "reservoirs", thanks to their large
magnetic fields which confine high-energy protons for $\sim 10^5$ years. Over
such long times, cosmic-ray transport can be significantly affected by
scatterings with sub-GeV dark matter. Here we point out that this scattering
distorts the cosmic-ray spectrum, and the distortion can be indirectly observed
by measuring the gamma-rays produced by cosmic-rays via hadronic collisions.
Present gamma-ray data show no sign of such a distortion, leading to stringent
bounds on the cross section between protons and dark matter. These are highly
complementary with current bounds and have large room for improvement with the
future gamma-ray measurements in the 0.1-10 TeV range from the Cherenkov
Telescope Array, which can strengthen the limits by as much as two orders of
magnitude.
