  We construct a new class of perturbative extremal charged rotating black hole
solutions in five dimensions in the Einstein--Born-Infeld theory. We start with
an extremal five-dimensional Myers-Perry black hole seed in which the two
possible angular momenta have equal magnitude, and add Born-Infeld electrical
charge keeping the extremality condition as a constraint. The perturbative
parameter is assumed to be the electric charge $q$ and the perturbations are
performed up to 4th order. We also study some physical properties of these
black holes. It is shown that the perturbative parameter $q$ and the
Born-Infeld parameter $\beta$ modify the values of the physical quantities of
the black holes. The solution suggests that the magnetic moment and
gyromagnetic ratio of the black hole spacetime change sign when the Born-Infeld
character of the solution starts to depart strongly from the Maxwell limit. We
venture an interpretation for the effect.
