We present new constraints on sub-GeV dark-matter particles scattering off electrons based on 6780.0 kg d of data collected with the DarkSide-50 dual-phase argon time projection chamber. This analysis uses electroluminescence signals due to ionized electrons extracted from the liquid argon target. The detector has a very high trigger probability for these signals, allowing for an analysis threshold of three extracted electrons, or approximately 0.05 keVee. We calculate the expected recoil spectra for dark matter-electron scattering in argon and, under the assumption of momentum-independent scattering, improve upon existing limits from XENON10 for dark-matter particles with masses between 30 and 100 MeV/c^{2}.