  We construct the first comprehensive radioactive background model for a dark
matter search with charge-coupled devices (CCDs). We leverage the
well-characterized depth and energy resolution of the DAMIC at SNOLAB detector
and a detailed GEANT4-based particle-transport simulation to model both bulk
and surface backgrounds from natural radioactivity down to 50 eV$_{\text{ee}}$.
We fit to the energy and depth distributions of the observed ionization events
to differentiate and constrain possible background sources, for example, bulk
$^{3}$H from silicon cosmogenic activation and surface $^{210}$Pb from radon
plate-out. We observe the bulk background rate of the DAMIC at SNOLAB CCDs to
be as low as $3.1 \pm 0.6$ counts kg$^{-1}$ day$^{-1}$ keV$_{\text{ee}}^{-1}$,
making it the most sensitive silicon dark matter detector. Finally, we discuss
the properties of a statistically significant excess of events over the
background model with energies below 200 eV$_{\text{ee}}$.
