If $X$ is a (general) scheme and $X$ is reduced at $p$, i.e. $\mathscr{O}_{X,p}$ is reduced, does there necessarily exist an open neighborhood of $p$ on which $X$ is reduced, i.e. $\mathscr{O}_X(U)$ is reduced?
This should be true at least when $X$ is Noetherian, since one can pick generators for the nilradical of any affine neighborhood $U$ and kill them off. How generally is it true?