  The island rule for the entanglement entropy is applied to an eternal
Reissner-Nordstr\"om black hole. The key ingredient is that the black hole is
assumed to be in thermal equilibrium with a heat bath of an arbitrary
temperature and so the generalized entropy is treated as being off-shell.
Taking the on-shell condition to the off-shell generalized entropy, we find the
generalized entropy and then obtain the entanglement entropy following the
island rule. For the non-extremal black hole, the entanglement entropy grows
linearly in time and can be saturated after the Page time as expected. The
entanglement entropy also has a well-defined Schwarzschild limit. In the
extremal black hole, the island prescription provides a logarithmically growing
entanglement entropy in time and a constant entanglement entropy after the Page
time. In the extremal black hole, the boundary of the island hits the curvature
singularity where the semi-classical approximations appear invalid. To avoid
encountering the curvature singularity, we apply this procedure to the Hayward
black hole regular at the origin. Consequently, the presence of the island in
extremal black holes can provide a finite entanglement entropy, which might
imply non-trivial vacuum configurations of extremal black holes.
