A novel mechanism to produce and detect light dark matter in experiments making use of GeV electrons (and positrons) impinging on a thick target (beam dump) is proposed. The positron-rich environment produced by the electromagnetic shower allows us to produce an A^{'} via nonresonant (e^{+}+e^{-}→γ+A^{'}) and resonant (e^{+}+e^{-}→A^{'}) annihilation on atomic electrons. The latter mechanism, for some selected kinematics, results in a larger sensitivity with respect to limits derived by the commonly used A^{'}-strahlung. This idea, applied to beam-dump experiments and active beam-dump experiments, pushes down the current limits by an order of magnitude.