Why do seemingly identical cells respond differently to a drug? To address this, we studied the dynamics and variability of the protein response of human cancer cells to a chemotherapy drug, camptothecin. We present a dynamic-proteomics approach that measures the levels and locations of nearly 1000 different endogenously tagged proteins in individual living cells at high temporal resolution. All cells show rapid translocation of proteins specific to the drug mechanism, including the drug target (topoisomerase-1), and slower, wide-ranging temporal waves of protein degradation and accumulation. However, the cells differ in the behavior of a subset of proteins. We identify proteins whose dynamics differ widely between cells, in a way that corresponds to the outcomes-cell death or survival. This opens the way to understanding molecular responses to drugs in individual cells.