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 $c_0$?

edit(2024-01-14): Let $\mathfrak{X}_{\omega_1}$ be the reflexive Banach space constructed by Argyros, Lopez-Abad, and Todorcevic [1, 2].

Q2: Is it known whether $ \mathfrak{X}_{\omega_1} \mathbin{\hat{\otimes}}_\pi \mathfrak{X}_{\omega_1}$ contains a copy of $c_0$?