In an ultrametric space, is every open set closed?

No. In the $p$-adic numbers $\Bbb Q_p$, one-point subsets such as $\{0\}$ are closed, but not open. The complement of a one-point subset is open, but not closed.

For a simple ad hoc example, take $\mathbb R$ or $\mathbb Q$ and define $d(x,y)=\max(|x|,|y|)$ for $x\ne y$; the set $\{0\}$ is closed but not open, so its complement is open but not closed.