The intracellular signalling network of the p53 protein plays important roles in various cellular processes, including the regulation of the cell cycle phase transitions, apoptosis, senescence, DNA repair and cell differentiation. These activities are determined by a variety of biochemical events such as transcriptional activation, post-transcriptional modifications (phosphorylation, ubiquitination, etc.), and they involve a large set of target genes and interacting proteins. Observed oscillatory behaviour in single cells under stress conditions inspires several research groups in simulating and studying the dynamics of the protein with the perspective of a proper understanding of physiological meanings of the oscillations. We propose a spatio-temporal PDE model of p53 activation and regulation in single cells following DNA damage, and we show that p53 oscillations can be retrieved by plainly involving p53-Mdm2 and ATM-p53-Wip1 negative feedbacks, which are sufficient for oscillations experimentally observed, with no further need to introduce delays in representing protein responses and without considering additional positive feedbacks. As observed in experiments in vivo, the ATM-p53-Wip1 negative feedback loop is essential while the coupled system p53-Mdm2 alone is not sufficient to retrieve oscillations. The oscillatory response of p53, as the outcome from the model, is sensitive to low DNA damage and the oscillations can be obtained for damage doses of arbitrary levels greater than a certain threshold (signalised by a supercritical Hopf point in the equilibrium curve), which is also in agreement with observations.