We study the electrodynamics of a superconductor superlattice consisting of layers of two kinds (wide layers with extremely weak pinning and very thin layers with-strong pinning), in a perpendicular magnetic field. The problem is treated with the nonlocality of the intervortex interaction and the Abrikosov-vortex elasticity simultaneously taken into account. The exact solution obtained for the surface impedance reveals a rather unusual behavior: By contrast with the monotonic dc magnetic-field dependence of the surface resistance rho(s) for a homogeneous superconductor, this dependence is non-monotonic monotonic (rho(s) is-proportional-to B for B almost-equal-to H(c1) and rho(s) is-proportional-to B-1/4 for B much greater than H(c1)) for the superconductor superlattice under study.