Norm-Euclidean integers of discriminant minus seven

The ring of integers $\mathbb{Z}[(1+\sqrt{-7})/2]$ is a Euclidean domain for $N(z)=z\overline{z}$. In its lattice choose the nearest row, whose imaginary-coordinate error is at most $\sqrt{7}/4$, then the nearest point on that row, whose real-coordinate error is at most $1/2$. Every complex number is thus within squared distance $11/16<1$ of a lattice point, giving Euclidean division. The suborder $\mathbb{Z}[\sqrt{-7}]$ has the nonprincipal ideal $(2,1+\sqrt{-7})$ and consequently is not a Euclidean domain for any Euclidean function.