Fibre product of smooth schemes over a singular scheme

Question

Let $k$ be a (perfect) field, let $f:X\to Z$ and $g:Y\to Z$ be morphisms of varieties (reduced separated scheme of finite type) over $k$, where $X$ and $Y$ are smooth over $k$, whereas $Z$ is not. Can we always show that the fibre product $X\times_Z Y$ is smooth over $k$? If so, does that rely on $Z$ being reduced, or can it be proven more generally? A reference would be appreciated.