  Cosmology in Eddington-inspired Born-Infeld gravity is investigated using a
scalar Born-Infeld field (e.g. tachyon condensate) as matter. In this way, both
in the gravity and matter sectors we have Born-Infeld-like structures
characterized by their actions and via two separate constants, $\kappa$ and
$\alpha_T^2$ respectively. With a particular choice of the form of $\dot{\phi}$
(the time derivative of the Born-Infeld scalar), analytical cosmological
solutions are found. Thereafter, we explore some of the unique features of the
corresponding cosmological spacetimes. For $\kappa>0$, our solution has a de
Sitter-like expansion both at early and late times, with an intermediate
deceleration sandwiched between the accelerating phases. On the other hand,
when $\kappa<0$, the initial de Sitter phase is replaced by a bounce. Our
solutions, at late time, fit well with available supernova data-- a fact we
demonstrate explicitly. The estimated properties of the Universe obtained from
the fitting of the $\kappa>0$ solution, are as good as in $\Lambda$CDM
cosmology. However, the $\kappa<0$ solution has to be discarded due to the
occurrence of a bounce at an unacceptably low redshift.
