The question is as in the title:

Is the ring $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_p = \mathbb{Z}_p \otimes_{\mathbb{Z}_{(p)}} \mathbb{Z}_p$ coherent?

As shown in the related question, the ring $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}_p$ *is* coherent, and so one cannot reduce to this case. (On the other hand $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}_p$ is not Noetherian, and so neither is $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_p$). The flatness lemma used in the answer to the rational case does also not appear to be applicable in this case.