Let $R$ be a complete discrete valuation ring with uniformizer $\pi$. Let $f:X\to \mathop{\mathrm{Spec}} R$ be a morphism of finite type and assume that the smooth locus of $f$ contains every $x\in X$ with $f(x)=(\pi)$. Is $f$ necessarily smooth?
Note that the assumption on the smooth locus is equivalent to the assumption that the base change of $f$ with respect to $R\to R/(\pi^n)$ is smooth for every $n\in\mathbb{N}$ (use Proposition 17.14.2 in EGA IV which states that it suffices to verify the infinitesimal lifting criterion with local Artin rings as test objects).