  All attempts to directly detect particle dark matter (DM) scattering on
nuclei suffer from the partial or total loss of sensitivity for DM masses in
the GeV range or below. We derive novel constraints from the inevitable
existence of a subdominant, but highly energetic, component of DM generated
through collisions with cosmic rays. Subsequent scattering inside conventional
DM detectors, as well as neutrino detectors sensitive to nuclear recoils,
limits the DM-nucleon scattering cross section to be below $10^{-31}$ cm$^2$
for both spin-independent and spin-dependent scattering of light DM.
