  We reconsider the problem of $f(R)$ theories of gravity coupled to
Born-Infeld theory of electrodynamics formulated in a Palatini approach, where
metric and connection are independent fields. By studying electrovacuum
configurations in a static and spherically symmetric space-time, we find
solutions which reduce to their Reissner-Nordstr\"om counterparts at large
distances but undergo important non-perturbative modifications close to the
center. Our new analysis reveals that the point-like singularity is replaced by
a finite-size wormhole structure, which provides a geodesically complete and
thus nonsingular space-time, despite the existence of curvature divergences at
the wormhole throat. Implications of these results, in particular for the
cosmic censorship conjecture, are discussed.
