We report direct-detection constraints on light dark matter particles interacting with electrons. The results are based on a method that exploits the extremely low levels of leakage current of the DAMIC detector at SNOLAB of 2-6×10^{-22} A cm^{-2}. We evaluate the charge distribution of pixels that collect <10e^{-} for contributions beyond the leakage current that may be attributed to dark matter interactions. Constraints are placed on so-far unexplored parameter space for dark matter masses between 0.6 and 100 MeV c^{-2}. We also present new constraints on hidden-photon dark matter with masses in the range 1.2-30 eV c^{-2}.