Understanding exotic, non-s-wave-like states of Cooper pairs is important and may lead to new superconductors with higher critical temperatures and novel properties. Their existence is known to be possible but has always been thought to be associated with non-traditional mechanisms of superconductivity where electronic correlations play an important role. Based on the first principles linear response calculation, we propose that an unconventional p-wave-like state can be favoured via a conventional phonon-mediated mechanism, as driven by an unusual, almost singular behaviour of the electron–phonon interaction at long wavelengths. Considering both harmonic and anharmonic contributions to electron-phonon coupling, we also predict that BiS2-family is a conventional superconductor near the charge-density-wave instability.