  Recently it was shown that the Born-Infeld-type modification of the quadratic
Yang-Mills action gives rise to classical particle-like solutions in the flat
space which have a striking similarity with the Bartnik-McKinnon solutions
known in the gravity coupled Yang-Mills theory. We show that both families are
continuously related within the framework of the Einstein-Born-Infeld theory
through interpolating sequences of parameters. We also investigate an internal
structure of the associated black holes. It is found that the Born-Infeld
non-linearity leads to a drastic modification of the black hole interior
typical for the usual Yang-Mills theory. In the latter case a generic solution
exhibits violent metric oscillations near the singularity. In the Born-Infeld
case a generic interior solution is smooth, the metric has the standard
Schwarzschild type singularity, and we did not observe internal horizons. Such
smoothing of the 'violent' EYM singularity may be interpreted as a result of
quantum effects.
