  We discuss prospects of searching for a dark photon ($A'$) which serves as
mediator between Standard model (SM) particles and light dark matter (LDM) by
using the combined results from the NA64 experiment at the CERN SPS running in
high-energy electron (NA64e) and muon (NA64$\mu$) modes. We discuss the most
natural values and upper bounds on the $A'$ coupling constant to LDM and show
they 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 and
Majorana LDM scenarios, the combined NA64e and NA64$\mu$ results with $\simeq
10^{13}$ EOT and a few $10^{13}$ MOT, respectively, will allow covering
significant region in the parameter space of the most interesting LDM models.
This makes NA64e and NA64$\mu$ extremely complementary to each other and
increases significantly the discovery potential of sub-GeV DM.
