  While non-rotating black-hole solutions are well known in Einstein--\ae{}ther
gravity, no axisymmetric solutions endowed with Killing horizons have been so
far found outside of the slowly rotating limit. Here we show that the Kerr
spacetime is also an exact vacuum solution of Einstein--\ae{}ther gravity in a
phenomenologically viable corner of the parameter space; the corresponding
\ae{}ther flow is characterised by a vanishing expansion. Such solution
displays all the characteristic features of the Kerr metric (inner and outer
horizons, ergoregion, etc.) with the remarkable exception of the
causality-violating region in proximity of the ring singularity. However, due
to the associated \ae{}ther flow, it is endowed with a special surface, inside
the Killing horizon, which exhibits many features normally related to the
universal horizon of the non-rotating solutions -- to which it tends in the
limit of zero angular momentum. Hence, these Kerr black holes are very good
mimickers of their general relativity counterparts while sporting important
differences and specific structures. As such, they appear particularly
well-suited candidates for future phenomenological studies.
