  We consider direct-detection searches for sub-GeV dark matter via electron
scatterings in the presence of large interactions between dark and ordinary
matter. Scatterings both on electrons and nuclei in the Earth's crust,
atmosphere, and shielding material attenuate the expected local dark matter
flux at a terrestrial detector, so that such experiments lose sensitivity to
dark matter above some critical cross section. We study various models,
including dark matter interacting with a heavy and ultralight dark photon,
through an electric dipole moment, and exclusively with electrons. For a
dark-photon mediator and an electric dipole interaction, the dark
matter-electron scattering cross-section is directly linked to the dark
matter-nucleus cross section, and nuclear interactions typically dominate the
attenuation process. We determine the exclusion bands for the different
dark-matter models from several experiments - SENSEI, CDMS-HVeV, XENON10,
XENON100, and DarkSide-50 - using a combination of Monte Carlo simulations and
analytic estimates. We also derive projected sensitivities for a detector
located at different depths and for a range of exposures, and calculate the
projected sensitivity for SENSEI at SNOLAB and DAMIC-M at Modane. Finally, we
discuss the reach to high cross sections and the modulation signature of a
small balloon- and satellite-borne detector sensitive to electron recoils, such
as a Skipper-CCD. Such a detector could potentially probe unconstrained
parameter space at high cross sections for a sub-dominant component of dark
matter interacting with a massive, but ultralight, dark photon.
