I know that an Euclidean ring is a principal ring and this last is a factorial ring.
I also know that if $B$ is a field, then $B[x]$ is Euclidean. While, for instance $\mathbb{Z}[x]$ is not a principal ring, while it is factorial (since $\mathbb{Z}$ is factorial).
Is there an example of a commutative ring with unity $R$ such that $R[x]$ is a principal ring but not an Euclidean one?
No, there is no such example, because if the polynomial ring $R[x]$ is a PID, then $R$ is a field, so that $R[x]$ is also Euclidean. For references see here.