  Employing higher order perturbation theory, we find a new class of
perturbative extremal rotating black hole solutions with Born-Infeld electric
charge in odd D-dimensional spacetime. The seed solution is an odd dimensional
extremal Myers-Perry black hole with equal angular momenta to which a
perturbative nonlinear electric Born-Infeld field charge q is added maintaining
the extremality condition. The perturbations are performed up to third order.
We also study some physical properties of these black holes. In particular, it
is shown that the values of the gyromagnetic ratio of the black holes are
modified by the perturbative parameter q and the Born-Infeld parameter {\beta}.
