Expansions of real closed fields with the Banach fixed point property

We study a variant of converses of the Banach fixed point theorem and its connection to tameness in expansions of a real closed field. An expansion of a real closed ordered field is said to have the Banach fixed point property when, for every locally closed definable set $E$, if every definable contraction on $E$ has a fixed point, then $E$ is closed. Let $R$ be an expansion of a real closed field. We prove that if $R$ has an o-minimal open core, then it has the Banach fixed point property; and if $R$ is definably complete and has the Banach fixed point property, then it has a locally o-minimal open core.