There exist Banach spaces $X$ such that the projective tensor product $X\mathbin{\hat{\otimes}}_\pi X$ contains an isomorphic copy of $c_0$ [BourgainPisier1983]. Moreover, $X$ is an $\mathcal{L}_\infty$-space with RNP (so contains no copies of $c_0$).
It was shown in [Pisier1983] that there exists a Banach space $X$ such that $ X \mathbin{\hat{\otimes}}_\pi X = X \mathbin{\hat{\otimes}}_\varepsilon X$, i.e. the twofold projective and injective tensor products are equal. The author had asked in Problem 4.5 in [Pisier1983] if there exists a reflexive Banach space with the same property.
Perhaps somewhat related question is:
Q: Does there exist a reflexive Banach space $Y$ such that the projective tensor product $ Y \mathbin{\hat{\otimes}}_\pi Y$ contains an isomorphic copy of $c_0$?
If exists, $Y$ doesn't have the approximation property (AP). Indeed, for $Y$ reflexive, the dual of the injective tensor product $(Y^*\mathbin{\hat{\otimes}_{\epsilon}}Y^*)^*$ has RNP by Theorem 1.9 in [RuessStegall1982]. If $Y$ has AP, it has metric AP (e.g. $\S$4.2 in in Ryan's book). So $Y\mathbin{\hat{\otimes}_{\pi}}Y$ is a closed subspace of $(Y^*\mathbin{\hat{\otimes}_{\epsilon}}Y^*)^*$ (e.g. Theorem 4.14 in Ryan's book), and so it cannot contain a copy of $c_0$.
\otimesbehaves in LaTeX and MathJax like a binary operation symbol, i.e. you see $X\otimes X$ and not $X{\otimes}X,$ but\hat{otimes}does not, i.e. you see $X\hat{\otimes}X$ and not $X\mathbin{\hat{\otimes}}X.$ I corrected that typo in this question by coding it asX\mathbin{\hat{\otimes}}X. $\qquad$ $\endgroup$