  We study static, spherically symmetric solutions with an electric field in an
extension of general relativity (GR) containing a Ricci-squared term and
formulated in the Palatini formalism. We find that all the solutions present a
central core whose area is proportional to the Planck area times the number of
charges. Far from the core, curvature invariants quickly tend to those of the
usual Reissner-Nordstr\"om solution, though the structure of horizons may be
different. In fact, besides the structures found in the Reissner-Nordstr\"om
solution of GR, we find black hole solutions with just one nondegenerate
horizon (Schwarzschild-like), and nonsingular black holes and naked cores. The
charge-to-mass ratio of the nonsingular solutions implies that the core matter
density is independent of the specific amounts of charge and mass and of order
the Planck density. We discuss the physical implications of these results for
astrophysical and microscopic black holes, construct the Penrose diagrams of
some illustrative cases, and show that the maximal analytical extension of the
nonsingular solutions implies a bounce of the radial coordinate.
