The interaction of heavy charged particles with DNA is of interest for hadrontherapy and the aerospace industry. Here, a time-dependent density functional theory study on the interaction of a 4 keV proton with an isolated DNA base pair (bp) was carried out. Ehrenfest dynamics was used to study the evolution of the system up to about 193 fs. It was observed that the dissociation of the target occurs between 80 and 100 fs. The effect of bp linking to the DNA double helix was emulated by fixing the four O3' atoms responsible for the attachment. The bp tends to dissociate into its main components, namely, the phosphate groups, sugars, and nitrogenous bases. A central impact with an energy transfer of 17.9 eV only produces a base damage while keeping the backbone intact. An impact on a phosphate group with an energy transfer of about 60 eV leads to a backbone break at that site together with a base damage, and the opposite backbone site integrity is kept. As the whole system is perturbed during this collision, no atom remains passive. These results suggest that base damage accompanies all backbone breaks as the hydrogen bonds that keep bases together are much weaker that those between the other components of the DNA.