  A perfect fluid, spatially flat cosmology in a $f(T)$ model, derived from a
recently proposed general Born-Infeld type theory of gravity is studied. Four
dimensional cosmological solutions are obtained assuming the equation of state
$p=\omega \rho$. For a positive value of $\lambda $ (a parameter in the theory)
the solution is singular (of big-bang type) but may have accelerated expansion
at an early stage. For $\lambda<0$ there exists a non-zero minimum scale factor
and a finite maximum value of the energy density, but the curvature scalar
diverges. Interestingly, for $\lambda <0$, the universe may undergo an eternal
accelerated expansion with a de Sitter expansion phase at late times. We find
these features without considering any extra matter field or even negative
pressure. Fitting our model with Supernova data we find that the simplest dust
model ($p=0$), with $\lambda >0$, is able to generate acceleration and fits
well, although the resulting properties of the universe differ much from the
known, present day, accepted values. The best fit model requires (with $\lambda
> 0$) an additional component of the physical matter density, with a negative
value of the equation of state parameter, along with dust. The $\lambda < 0$
solutions do not fit well with observations. Though these models do not explain
the dark energy problem with consistency, their analysis does shed light on the
plausibility of an alternative geometrical explanation.
