  We study Born-Infeld gravity coupled to an anisotropic fluid in a static,
spherically symmetric background. The free function characterizing the fluid is
selected on the following grounds: i) recovery of the Reissner-Nordstr\"om
solution of GR at large distances, ii) fulfillment of classical energy
conditions and iii) inclusion of models of nonlinear electrodynamics as
particular examples. Four branches of solutions are obtained, depending on the
signs of two parameters on the gravity and matter sectors. On each branch, we
discuss in detail the modifications on the innermost region of the
corresponding solutions, which provides a plethora of configurations, including
nonsingular black holes and naked objects, wormholes and de Sitter cores. The
regular character of these configurations is discussed according to the
completeness of geodesics and the behaviour of curvature scalars.

General Relativity and Quantum Cosmology;High Energy Physics - Theory