Tumour evolution is driven by both genetic and epigenetic changes. CENP-A, the centromeric histone H3 variant, is an epigenetic mark that directly perturbs genetic stability and chromatin when overexpressed. Although CENP-A overexpression is a common feature of many cancers, how this impacts cell fate and response to therapy remains unclear. Here, we established a tunable system of inducible and reversible CENP-A overexpression combined with a switch in p53 status in human cell lines. Through clonogenic survival assays, single-cell RNA-sequencing and cell trajectory analysis, we uncover the tumour suppressor p53 as a key determinant of how CENP-A impacts cell state, cell identity and therapeutic response. If p53 is functional, CENP-A overexpression promotes senescence and radiosensitivity. Surprisingly, when we inactivate p53, CENP-A overexpression instead promotes epithelial-mesenchymal transition, an essential process in mammalian development but also a precursor for tumour cell invasion and metastasis. Thus, we uncover an unanticipated function of CENP-A overexpression to promote cell fate reprogramming, with important implications for development and tumour evolution.