The DAMIC-M experiment employs thick, fully depleted silicon charge-coupled devices (CCDs) to search for sub-GeV dark matter particles. Thanks to its multiple non-destructive measurements of the pixel charge, DAMIC-M skipper CCDs achieve single-ionization charge resolution and an energy threshold in the eV-scale. We report on the progress of the experiment and first results from a prototype detector installed underground at the Laboratoire Souterrain de Modane. In particular, constraints on dark matter particles interacting with electrons are obtained in a mass range between 0.5 and 1000 MeV/c$^2$. We also present results of a search for diurnal modulation in the measured single-ionization charge rate which significantly improves sensitivity at the lowest masses.