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, $��$ and $��_T^2$ respectively. With a particular choice of the form of $\dot��$ (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 $��>0$, 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 $��<0$, 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 $��>0$ solution, are as good as in $��$CDM cosmology. However, the $��<0$ solution has to be discarded due to the occurrence of a bounce at an unacceptably low redshift.

24 pages, 8 figures. Published version