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][1]]. Moreover, $X$ is an $\mathcal{L}_\infty$-space with RNP (so contains no copies of $c_0$).

It was shown in [[Pisier1983][2]] 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][2]] 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][3]]. If $Y$ has AP, it has metric AP (e.g. [$\S$4.2][4] 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][5] in Ryan's book), and so it cannot contain a copy of $c_0$. 


  [1]: https://doi.org/10.1007/BF02584862
  [2]: https://doi.org/10.1007/BF02393206
  [3]: https://doi.org/10.1007/BF01457455
  [4]: https://books.google.ca/books?id=MSLUBwAAQBAJ&pg=PA82
  [5]: https://books.google.ca/books?id=MSLUBwAAQBAJ&pg=PA80