Recent observations show that the single-cell response of p53 to ionizing radiation (IR) is "digital" in that it is the number of oscillations rather than the amplitude of p53 that shows dependence on the radiation dose. We present a model of this phenomenon. In our model, double-strand break (DSB) sites induced by IR interact with a limiting pool of DNA repair proteins, forming DSB-protein complexes at DNA damage foci. The persisting complexes are sensed by ataxia telangiectasia mutated (ATM), a protein kinase that activates p53 once it is phosphorylated by DNA damage. The ATM-sensing module switches on or off the downstream p53 oscillator, consisting of a feedback loop formed by p53 and its negative regulator, Mdm2. In agreement with experiments, our simulations show that by assuming stochasticity in the initial number of DSBs and the DNA repair process, p53 and Mdm2 exhibit a coordinated oscillatory dynamics upon IR stimulation in single cells, with a stochastic number of oscillations whose mean increases with IR dose. The damped oscillations previously observed in cell populations can be explained as the aggregate behavior of single cells.