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We discuss prospects of searching for a dark photon ($A'$) which serves asmediator between Standard model (SM) particles and light dark matter (LDM) byusing the combined results from the NA64 experiment at the CERN SPS running inhigh-energy electron (NA64e) and muon (NA64$\mu$) modes. We discuss the mostnatural values and upper bounds on the $A'$ coupling constant to LDM and showthey are lying in the range accessible at NA64. While for the projected $5\times10^{12}$ electrons on target (EOT) NA64e is able to probe the scalar andMajorana LDM scenarios, the combined NA64e and NA64$\mu$ results with $\simeq10^{13}$ EOT and a few $10^{13}$ MOT, respectively, will allow coveringsignificant region in the parameter space of the most interesting LDM models.This makes NA64e and NA64$\mu$ extremely complementary to each other andincreases significantly the discovery potential of sub-GeV DM.

Combined search for light dark matter with electron and muon beams at NA64