  We consider electrovacuum black hole spacetimes in classical extensions of
Eddington-inspired Born-Infeld gravity. By rewriting Born-Infeld action as the
square root of the determinant of a matrix $\hat{\Omega}$, we consider the
family of models $f (|\hat{\Omega}|)$, and study black hole solutions for a
power-law family of models labelled by a simple parameter. We show how the
innermost structure of the corresponding black holes is modified as compared to
their General Relativity counterparts, discussing in which cases a wormhole
structure replaces the point-like singularity. We go forward to argue that in
such cases a geodesically complete and thus non-singular spacetime is present,
despite the existence of curvature divergences at the wormhole throat.
