The ring of integers $\mathcal{O}_{\mathbf{C}_p}$ of $\mathbf{C}_p$ is not noetherian, but its only localization is $\mathbf{C}_p$, which is noetherian.

The point here is that the valuation of $\mathcal{C}_p$ takes values in $\mathbf{Q}$, so the maximal ideal of its valuation ring cannot be finitely generated.