We present measurements of bulk radiocontaminants in the high-resistivity silicon CCDs from the DAMIC at SNOLAB experiment. We utilize the exquisite spatial resolution of CCDs to discriminate between $alpha$ and $beta$ decays, and to search with high efficiency for the spatially-correlated decays of various radioisotope sequences. Using spatially-correlated $beta$ decays, we measure a bulk radioactive contamination of $^{32}$Si in the CCDs of $140 pm 30$ $mu$Bq/kg, and place an upper limit on bulk $^{210}$Pb of $< 160~mu$Bq/kg. Using similar analyses of spatially-correlated bulk $alpha$ decays, we set limits of $< 11$ $mu$Bq/kg (0.9 ppt) on $^{238}$U and of $< 7.3$ $mu$Bq/kg (1.8 ppt) on $^{232}$Th. The ability of DAMIC CCDs to identify and reject spatially-coincident backgrounds, particularly from $^{32}$Si, has significant implications for the next generation of silicon-based dark matter experiments, where $beta$s from $^{32}$Si decay will likely be a dominant background. This capability demonstrates the readiness of the CCD technology to achieve kg-scale dark matter sensitivity.
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