Take an integral, proper variety $X$ over $k$ with function field $k(X)$. Let $A$ be a DVR containing $k$ having field of fractions $k(X)$. Take $P \in X$. Does there always exist an injection $\mathcal{O}_{X,P} \to A$?
EDIT: I changed the question a very little bit (replaced $\subseteq$ by the existence of an injection).