  We construct a new class of charged rotating solutions of $(n+1)$-dimensional
Einstein-Born-Infeld gravity with cylindrical or toroidal horizons in the
presence of cosmological constant and investigate their properties. These
solutions are asymptotically (anti)-de Sitter and reduce to the solutions of
Einstein-Maxwell gravity as the Born-Infeld parameters goes to infinity. We
find that these solutions can represent black branes, with inner and outer
event horizons, an extreme black brane or a naked singularity provided the
parameters of the solutions are chosen suitably. We compute temperature, mass,
angular momentum, entropy, charge and electric potential of the black brane
solutions. We obtain a Smarr-type formula and show that these quantities
satisfy the first law of thermodynamics. We also perform a stability analysis
by computing the heat capacity and the determinant of Hessian matrix of mass
with respect to its thermodynamic variables in both the canonical and the
grand-canonical ensembles, and show that the system is thermally stable in the
whole phase space.
