Integral domain whose irreducible elements are not prime

I found this paper which in Example 2.2 (d) (page 4) claims that $K[[X^2,X^3]]$ is an integral domain with no prime elements but many irreducible elements, for any field $K$.

Hint $ $ The irreducibles of $\,\Bbb Q+x\Bbb R[[x]]\,$ are $\,rx\ne 0,\,r\in \Bbb R.\,$ But $\,rx\mid (\pi rx)^2,\,\ rx\nmid \pi r x\,$ by $\,\pi\not\in\Bbb Q$

Unfortunately I do not have the rep to comment on your answer above, but I just wanted to point out that this example is very similar to another interesting example of factorization properties.

Looking at $R=\mathbb{R}[X^2,X^3]$. It is atomic in the sense that every non-zero non-unit has a factorization into irreducibles, but the lengths of these factorizations can vary. This would be impossible for a factorization into primes. In a domain, if an element has two prime factorizations, they must be unique up to rearrangement and associate.

Consider the factorizations of the element $X^6=X^2\cdot X^2 \cdot X^2 = X^3 \cdot X^3$ are two factorizations into irreducibles. The first of length 3 and the second of length 2. ($X^3$ and $X^2$ are irreducible since $X \notin R$). This is an example of a bounded factorization domain which fails to be a half factorization domain.