The contact problem of an elastic sphere indenting an elastic-perfectly plastic substrate having an elastic layer is analysed using the finite element method and compared with a purely elastic analytical solution. The case of a layer stiffer and harder than the substrate is investigated and solutions for the contact pressure, subsurface stresses and strains are presented for various indentation depths. The results show good agreement until the threshold of plasticity is reached. Thereafter, as the system is subjected to higher penetrations, the two solutions diverge as plastic deformation initiates in the finite element model. The analysis also demonstrates that the influence of a deformable indenter upon the plastic solutions cannot be neglected.