Dark matter (DM) particles with sufficiently large cross sections may scatter as they travel through Earth's bulk. The corresponding changes in the DM flux give rise to a characteristic daily modulation signal in detectors sensitive to DM-electron interactions. Here, we report results obtained from the first underground operation of the DAMIC-M prototype detector searching for such a signal from DM with MeV-scale mass. A model-independent analysis finds no modulation in the rate of 1 e^{-} events with sidereal period, where a DM signal would appear. We then use these data to place exclusion limits on DM in the mass range [0.53,2.7] MeV/c^{2} interacting with electrons via a dark photon mediator. Taking advantage of the time-dependent signal we improve by ∼2 orders of magnitude on our previous limit obtained from the total rate of 1 e^{-} events, using the same dataset. This daily modulation search represents the current strongest limit on DM-electron scattering via ultralight mediators for DM masses around 1 MeV/c^{2}.