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

It was shown in [[Pisier1983][1]] 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][1]] 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 $c_0$?



  [1]: https://doi.org/10.1007/BF02393206
  [2]: https://doi.org/10.1007/BF02584862