In a (necessarily non-Noetherian) integral domain $A$ of (Krull) dimension $1$, is it possible that there is an infinite collection of prime ideals $\mathfrak{p}_i$ such that $\cap_i \mathfrak{p}_i \neq 0$?

Context: part of Problem 5.2.4 in Qing Liu's *Algebraic Geometry and Arithmetic Curves* asks you to show that (after reducing to the ring statement) that for an affine integral scheme $X \simeq \text{Spec } A$ of dimension $1$, an element $\frac{p}{q}$ of $K(X)$ is contained in $\mathcal{O}_{X,x}$ for all but finitely many $x$. This means $q\neq 0$ is contained in only finitely many prime ideals.