  An important source of background in direct searches for low-mass dark matter
particles are the energy deposits by small-angle scattering of environmental
$\gamma$ rays. We report detailed measurements of low-energy spectra from
Compton scattering of $\gamma$ rays in the bulk silicon of a charge-coupled
device (CCD). Electron recoils produced by $\gamma$ rays from $^{57}$Co and
$^{241}$Am radioactive sources are measured between 60 eV and 4 keV. The
observed spectra agree qualitatively with theoretical predictions, and
characteristic spectral features associated with the atomic structure of the
silicon target are accurately measured for the first time. A
theoretically-motivated parametrization of the data that describes the Compton
spectrum at low energies for any incident $\gamma$-ray flux is derived. The
result is directly applicable to background estimations for low-mass dark
matter direct-detection experiments based on silicon detectors, in particular
for the DAMIC experiment down to its current energy threshold.
