Positrons beam dump experiments have unique features to search for very narrow resonances coupled superweakly to $e^+ e^-$ pairs. Due to the continue 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 $e^-$ in the target. In the case of a dark photon $A$ kinetically mixed with the photon, this production mode is of first order in the electromagnetic coupling $alpha$, and thus parametrically enhanced with respect to the $O(alpha^2)$ $e^+e^- to gamma A$ production mode and to the $O(alpha^3)$ $A$ bremsstrahlung in $e^--$nucleon scattering so far considered. If the lifetime is sufficiently long to allow the $A$ to exit the dump, $A to e^+e^-$ 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 $^8$Be anomaly in nuclear transitions.
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