The diffuse interstellar bands (DIBs) are absorption features observed in optical and near-infrared spectra that are thought to be associated with carbon-rich polyatomic molecules in interstellar gas. However, because the central wavelengths of these bands do not correspond with electronic transitions of any known atomic or molecular species, their nature has remained uncertain since their discovery almost a century ago. Here we report on unusually strong DIBs in optical spectra of the broad-lined Type Ic supernova SN 2012ap that exhibit changes in equivalent width over short (~30 days) timescales. The 4428 and 6283 Angstrom DIB features get weaker with time, whereas the 5780 Angstrom feature shows a marginal increase. These nonuniform changes suggest that the supernova is interacting with a nearby source of the DIBs and that the DIB carriers possess high ionization potentials, such as small cations or charged fullerenes. We conclude that moderate-resolution spectra of supernovae with DIB absorptions obtained within weeks of outburst could reveal unique information about the mass-loss environment of their progenitor systems and provide new constraints on the properties of DIB carriers.