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.