We present new direct-detection constraints on eV-to-GeV dark matter interacting with electrons using a prototype detector of the Sub-Electron-Noise Skipper-CCD Experimental Instrument. The results are based on data taken in the MINOS cavern at the Fermi National Accelerator Laboratory. We focus on data obtained with two distinct readout strategies. For the first strategy, we read out the Skipper CCD continuously, accumulating an exposure of 0.177 g day. While we observe no events containing three or more electrons, we find a large one- and two-electron background event rate, which we attribute to spurious events induced by the amplifier in the Skipper-CCD readout stage. For the second strategy, we take five sets of data in which we switch off all amplifiers while exposing the Skipper CCD for 120 ks, and then read out the data through the best prototype amplifier. We find a one-electron event rate of (3.51±0.10)×10-3 events/pixel/day, which is almost 2 orders of magnitude lower than the one-electron event rate observed in the continuous-readout data, and a two-electron event rate of (3.18-0.55+0.86)×10-5 events/pixel/day. We again observe no events containing three or more electrons, for an exposure of 0.069 g day. We use these data to derive world-leading constraints on dark matter-electron scattering for masses between 500 keV and 5 MeV, and on dark-photon dark matter being absorbed by electrons for a range of masses below 12.4 eV.