  We obtain new solutions of Einsteinian cubic gravity coupled to a Maxwell
field that describe the near-horizon geometry of charged and rotating black
holes. We show that the AdS$_2\times\mathbb{S}^2$ near-horizon geometry of
Reissner-Nordstr\"om black holes receives no corrections, but deviations with
respect to the extremal Kerr-Newman solution appear as we turn on the angular
momentum. We construct the profile of these corrected geometries using both
numerical methods and slowly-spinning expansions, but we also find additional
solutions that do not reduce to AdS$_2\times\mathbb{S}^2$ geometries in any
limit and that do not have a counterpart in Einstein gravity. Quite remarkably,
we are able to obtain closed-form exact expressions for the area and Wald's
entropy of all of these solutions, and using this result, we analyze the phase
space of extremal back holes in this theory. To the best of our knowledge, this
is the first time the entropy of a rotating black hole in higher-order gravity
has been exactly computed.
