  We study the nonlinear stability of the Cauchy horizon in the interior of
extremal Reissner-Nordstr\"om black holes under spherical symmetry. We consider
the Einstein-Maxwell-Klein-Gordon system such that the charge of the scalar
field is appropriately small in terms of the mass of the background extremal
Reissner-Nordstr\"om black hole. Given spherically symmetric characteristic
initial data which approach the event horizon of extremal Reissner-Nordstr\"om
sufficiently fast, we prove that the solution extends beyond the Cauchy horizon
in $C^{0,\frac{1}{2}}\cap W^{1,2}_{loc}$, in contrast to the subextremal case
(where generically the solution is $C^0\setminus (C^{0,\frac{1}{2}}\cap
W^{1,2}_{loc}))$. In particular, there exist non-unique spherically symmetric
extensions which are moreover solutions to the Einstein-Maxwell-Klein-Gordon
system. Finally, in the case that the scalar field is chargeless and massless,
we additionally show that the extension can be chosen so that the scalar field
remains Lipschitz.
