  Exact solutions describing rotating black holes can offer important tests for
alternative theories of gravity, motivated by the dark energy and dark matter
problems. We present an analytic rotating black hole solution for a class of
vector-tensor theories of modified gravity, valid for arbitrary values of the
rotation parameter. The new configuration is characterised by parametrically
large deviations from the Kerr-Newman geometry, controlled by non-minimal
couplings between vectors and gravity. It has an oblate horizon in
Boyer-Lindquist coordinates, and it can rotate more rapidly and have a larger
ergosphere than black holes in General Relativity (GR) with the same asymptotic
properties. We analytically investigate the features of the innermost stable
circular orbits for massive objects on the equatorial plane, and show that
stable orbits lie further away from the black hole horizon with respect to
rotating black holes in GR. We also comment on possible applications of our
findings for the extraction of rotational energy from the black hole.
