  The neutron-proton mass difference in finite nuclei is studied in the
framework of a medium-modified Skyrme model. The possible interplay between the
effective nucleon mass in finite nuclei and the Nolen-Schiffer anomaly is
discussed. In particular, we find that a correct description of the properties
of mirror nuclei leads to a stringent restriction of possible modifications of
the nucleon's effective mass in nuclei.
