Is a square of a prime ideal in a UFD always primary?

Question

More concretely,

Let $R$ be a UFD and $\mathfrak{p}$ a prime ideal ideal of $R$. Does it always hold that $\mathfrak{p}^2$ is a primary ideal?

I know that it always holds if $\mathfrak{p}$ is a principal ideal or a maximal ideal, so one needs only consider rings of Krull dimension $\geqslant 3$.

I proposed this question mainly because I'd like to know how well-behaved a prime ideal of UFD could be.

Answer 1

You can find (counter)examples even in polynomial rings over a field; see here.