  The existence of static, spherically symmetric, self-gravitating scalar field
solutions in the context of Born-Infeld gravity is explored. Upon a combination
of analytical approximations and numerical methods, the equations for a free
scalar field (without a potential term) are solved, verifying that the
solutions recover the predictions of General Relativity far from the center but
finding important new effects in the central regions. We find two classes of
objects depending on the ratio between the Schwarzschild radius and a length
scale associated to the Born-Infeld theory: massive solutions have a wormhole
structure, with their throat at $r\approx 2M$, while for the lighter
configurations the topology is Euclidean. The total energy density of these
solutions exhibits a solitonic profile with a maximum peaked away from the
center, and located at the throat whenever a wormhole exists. The geodesic
structure and curvature invariants are analyzed for the various configurations
considered.
