We propose to search for light $U(1)$ dark photons, $A$, produced via kinetically mixing with ordinary photons via the Compton-like process, $gamma e^- rightarrow A e^-$, in a nuclear reactor and detected by their interactions with the material in the active volumes of reactor neutrino experiments. We derive 95% confidence-level upper limits on $epsilon$, the $A$-$gamma$ mixing parameter, $epsilon$, for dark-photon masses below 1$sim$MeV of $epsilon~< ~1.3times 10^{-5}$ and $epsilon~<~2.1times 10^{-5}$, from NEOS and TEXONO experimental data, respectively. This study demonstrates the applicability of nuclear reactors as potential sources of intense fluxes of low-mass dark photons.