If $X$ is a (reduced) scheme and $P$ is a point of $X$ (not necessarily closed) such that the local ring $\mathcal{O}_{X,P}$ is a regular domain, then must there exist an open affine neighborhood $U = \text{Spec }A$ of $P$ such that $A$ is an integral domain?
I'm almost certain this is true, since the local ring being regular means that it doesn't sit in the intersection of irreducible components, and hence it must be "locally" irreducible...but I can't think how to prove it.
If needed, we can assume $X$ is also Noetherian.