  In this paper we study under which conditions the Reissner-Nordstr\"om-anti
de Sitter black hole can be a solution of the vacuum mimetic $F(R)$ gravity
with Lagrange multiplier and mimetic scalar potential. As we demonstrate, the
resulting picture in the mimetic $F(R)$ gravity case, is different in
comparison to the ordinary $F(R)$ gravity case, with the two descriptions
resulting to a different set of constraints that need to hold true. We also
investigate the metric perturbations in the mimetic $F(R)$ gravity case, for
the Reissner-Nordstr\"om-anti de Sitter black hole metric, at first order of
the perturbed variables. Interestingly enough, the resulting equations are
identical to the ones corresponding to the ordinary $F(R)$ gravity
Reissner-Nordstr\"om-anti de Sitter black hole, at least at first order. We
attribute this feature to the particular form of the Reissner-Nordstr\"om-anti
de Sitter metric, and we speculate for which cases there could be differences
between the mimetic and non-mimetic case. Since the perturbation equations are
the same for the two cases, it is possible to have black hole instabilities in
the mimetic $F(R)$ gravity case too, which can be interpreted as
anti-evaporation of the black hole.
