Positrons beam dump experiments have unique features to search for very narrow resonances coupled superweakly to ${e}^{+}{e}^{\ensuremath{-}}$ pairs. Due to the continued loss of energy from soft photon bremsstrahlung, in the first few radiation lengths of the dump a positron beam can continuously scan for resonant production of new resonances via ${e}^{+}$ annihilation off an atomic ${e}^{\ensuremath{-}}$ in the target. In the case of a dark photon ${A}^{\ensuremath{'}}$ kinetically mixed with the photon, this production mode is of first order in the electromagnetic coupling $\ensuremath{\alpha}$, and thus parametrically enhanced with respect to the $O({\ensuremath{\alpha}}^{2}){e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\gamma}{A}^{\ensuremath{'}}$ production mode and to the $O({\ensuremath{\alpha}}^{3}){A}^{\ensuremath{'}}$ bremsstrahlung in ${e}^{\ensuremath{-}}$-nucleon scattering so far considered. If the lifetime is sufficiently long to allow the ${A}^{\ensuremath{'}}$ to exit the dump, ${A}^{\ensuremath{'}}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}$ decays could be easily detected and distinguished from backgrounds. We explore the foreseeable sensitivity of the Frascati PADME experiment in searching with this technique for the 17 MeV dark photon invoked to explain the $^{8}\mathrm{Be}$ anomaly in nuclear transitions.