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.