Principal ideal domain not euclidean

Can anyone give an example of a principal ideal domain that is not Euclidean and is not isomorphic to $\mathbb{Z}[\frac{1+\sqrt{-a}}{2}]$, $a = 19,43,67,163$?

I believe it is conjectured that no other integer rings of number fields have this property. What about other rings?

One such simple example of a non-Euclidean PID is $ K[[x,y]][1/(x^2\!+\!y^3)]\,$ over any field $\,K,\,$ i.e. adjoin the inverse of $\,x^2\!+\!y^3$ to a bivariate power series ring over a field. For the proof, and a general construction method see

D.D. Anderson. An existence theorem for non-euclidean PID’s,
Communications in Algebra, 16:6, 1221-1229, 1988.

For number rings, by Weinberger (1973), assuming GRH, a UFD number ring R with infinitely many units is Euclidean, e.g. real quadratic number rings are Euclidean $\!\iff\!$ PID $\!\iff\!$ UFD.