To add to Achim's answer, Bousfield and Kan, "The core of a ring" JPAA 1972 and (correction) 1973 define a ring $R$ to be a "solid" if the multiplication map $R\otimes_{\mathbb{Z}}R\to R$ is an isomorphism. They then go on to characterise solid rings. There is also a related discussion on page 44 of their book, "Homotopy limits, completions and localizations".

To add to Achim's answer, Bousfield and Kan, "The core of a ring" JPAA 1972 and (correction) 1973 define a ring $R$ to be a "solid" if the multiplication map $R\otimes_{\mathbb{Z}}R\to R$ is an isomorphism. They then go on to characterise solid rings. There is also a related discussion on page 44 of their book, "Homotopy limits, completions and localizations".

If a ring $R$ is solid, then for any right $R$-module $A$ and left $R$-module $B$, we have $$A\otimes_{\mathbb{Z}} B \cong A\otimes_R R\otimes_{\mathbb{Z}} R \otimes_R B \cong A\otimes_R R \otimes_R B\cong A\otimes_R B.$$ So your question is asking whether the ring of $p$-adic numbers $\mathbb{Z}_p$ is solid.

The ring of $\mathbb{Z}_p$ is not solid in the sense of Bousfield and Kan. This follows from their paper, or from the answer of Achim above.