There exist Banach spaces $X$ such that the projective tensor product $\displaystyle X\hat{\otimes}_{\pi} X$ 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 $\displaystyle X\hat{\otimes}_{\pi} X = \displaystyle X\hat{\otimes}_{\epsilon} X$, 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 $\displaystyle Y\hat{\otimes}_{\pi} Y$ contains an isomorphic copy of $c_0$?