Calculations for infinite nuclear matter with realistic nucleon-nucleon interactions suggest that the isoscalar effective mass of a nucleon at the saturation density m * /m equals 0.8 ± 0.1. This result is at variance with empirical data on the level density in finite nuclei, which are consistent with m * /m ≈ 1. Ma and Wambach suggested that these two contradicting results may be reconciled within a single theoretical framework by assuming a radial-dependent effective mass, peaked at the nuclear surface. The aim of this exploratory work is to investigate this idea within the density functional theory by using a Skyrme-type local functional enriched with new terms, τ (∇ρ) 2 and τ (dρ/dr), where τ and ρ denote the kinetic and particle densities, respectively. We show that each of these terms can give rise to a surface peak in the effective mass, but of a limited height. We investigate the influence of the radial profile of the effective mass on the spin-orbit splittings and centroids. In particular, we demonstrate that the τ (dρ/dr) term quenches the 1f 5/2 -1f 7/2 splitting in 40 Ca, which is strongly overestimated within conventional Skyrme parametrizations.