  The electrically charged Born-Infeld black holes in the Palatini formalism
for $f(R)$ theories are analyzed. Specifically we study those supported by a
theory $f(R)=R\pm R^2/R_P$, where $R_P$ is Planck's curvature. These black
holes only differ from their General Relativity counterparts very close to the
center, but may give rise to different geometrical structures in terms of inner
horizons. The nature and strength of the central singularities are also
significantly affected. In particular, for the model $f(R)=R - R^2/R_P$ the
singularity is shifted to a finite radius, $r_+$, and the Kretschmann scalar
diverges only as $1/(r-r_+)^{2}$.
