The only example I know of occurs in On Hausdorff completions of commutative rings in rigid geometry by Fujiwara, Gabber, Kato (and according to the intro the example is due to Gabber). The example is the subject of $\S$8.3, in their language they give an example of a $t$-adically complete ring $A_0$ which is Noetherian outside $(t)$ but $A_0\langle x\rangle$ is not noetherian outside $(t)$. In this setting $A:=A_0[1/t]$ has a unique structure of a Tate-Huber ring and $A$ being Noetherian is the same as $A_0$ being Noetherian outside $(t)$, so one gets a Noetherian Tate ring $A$ for which $A\langle x\rangle$ is not Noetherian. In addition they prove in Lemma 8.3.3 that $A$ is in fact a field (but of course not a nonarchimedean field).

The only example I know of occurs in On Hausdorff completions of commutative rings in rigid geometry by Fujiwara, Gabber, Kato (and according to the intro the example is due to Gabber). The example is the subject of $\S$8.3, in their language they give an example of a $t$-adically complete ring $A_0$ which is Noetherian outside $(t)$ but $A_0\langle x\rangle$ is not noetherian outside $(t)$. In this setting $A:=A_0[1/t]$ has a unique structure of a Tate-Huber ring making $A_0$ a ring of definition, and then $A$ being Noetherian is the same as $A_0$ being Noetherian outside $(t)$, so one gets a Noetherian Tate ring $A$ for which $A\langle x\rangle$ is not Noetherian. In addition they prove in Lemma 8.3.3 that $A$ is in fact a field (but of course not a nonarchimedean field).