We study the electrodynamics of a superlattice that consists of two kinds of layers: the layers of a type-II superconductor and the layers of a normal metal in a perpendicular magnetic field. The problem is treated in the framework of London electrodynamics taking into account simultaneously the nonlocality of the intervortex interaction and the Abrikosov vortex elasticity. The dependence of the surface impedance Z on the dc-magnetic-field induction B much less than H(c2) (H(c2) is the upper critical field) is analzyed for different values of the parameters of the superlattice. It is shown that, in the low-frequency limit, three types of Z (B) dependence are possible: (i) Z is-proportional-to B for small fields and Z is-proportional-to B1/2 for large fields, with a crossover at B almost-equal-to H(c1) (H(c1) is the lower critical field); (ii) Z is-proportional-to B1/2 for any fields; or (iii) Z = AB1/2 for small fields and Z = CB1/2 for large fields, the coefficients A and C being nonequal, with the crossover AB1/2-->CB1/2 at B almost-equal-to H(c1). The type-(i) dependence characteristic of a uniform superconductor converts into a type-(ii) dependence as one diminishes the thickness of the superconductor layers. The physical origin of this conversion is explained. It is suggested that observed dependence in multilayer superconductor systems Z is-proportional-to B1/2 is due to the effect discussed in the paper.