If $R=\mathbb{Z}[\frac{1}{2}(1+\sqrt{-19})]$ is an example of a PID which is not a Euclidean domain. But since Euclidean domains may not necessarily be fields, this doesn't say much unless $R$ in this case or in other cases is a field. Is $R$ a field? What are ways to check if a PID is a field?
What are some examples of PIDs that are not fields?