preservation of localness among certain Krull domains

Question

Let $R$ be a local Krull domain, and let $\mathfrak p$ be a height one prime ideal whose class in the divisor class group is non-torsion. (That is, $\mathfrak p^{(n)}$ is non-principal for all $n$.) Obviously, this limits what $R$ can be -- e.g. it can't be one-dimensional. Let $$ S := \bigcap_{\mathfrak q \in Spec R, \text{ht}(\mathfrak q)=1, \mathfrak q \neq \mathfrak p} R_{\mathfrak q} $$ Is $S$ local? And if not, is $\mathfrak p S$ a proper ideal of $S$?