Superreflexive tensor product spaces

The aim of this note is to prove that, given two superreflexive Banach spaces $X$ and $Y$, then $X\widehat{\otimes}_\pi Y$ is superreflexive if and only if either $X$ or $Y$ is finite-dimensional. In a similar way, we prove that $X\widehat{\otimes}_\varepsilon Y$ is superreflexive if and only if either $X$ or $Y$ is finite-dimensional.