Magnetic screening in the proximity system ${S}_{\ensuremath{\alpha}}$-${M}_{\ensuremath{\beta}}$, where ${M}_{\ensuremath{\beta}}$ is a normal metal N, semiconductor (semimetal), or a superconductor, is studied. Main attention is paid to the low-temperature region where nonlocality plays an important role. The thermodynamic Green's-function method is employed in order to describe the behavior of the proximity system in an external field. The temperature and thickness dependences of the penetration depth \ensuremath{\lambda} are obtained. The dependence \ensuremath{\lambda}(T) differs in a striking way from the dependence in usual superconductors. The strong-coupling effect is taken into account. A special case of screening in a superconducting film backed by a size-quantizing semimetal film is considered. The results obtained are in good agreement with experimental data.