Gcd not implies Bezout

Statement
There exist gcd domains which are not Bezout domains.

Proof
Since every UFD is a gcd domain, it suffices to construct an example of a UFD which is not a Bezout domain. In particular, it suffices to construct an example of a Noetherian UFD which is not a PID. One such example is the multivariate polynomial ring over a field, i.e. the polynomial ring over a field, in more than one variable.