If an element in an integral domain is factorisable into irreducibles, then are all of its divisors factorisable into irreducibles as well?

Question

Let $D$ be an integral domain and let $x\in D$ be an element that can be written as a (finite) product of irreducible elements of $D$. Let $y$ be a non-unit divisor of $x$. Is it true that $y$ can be written as a product of irreducible elements as well?