In this paper we analyse when every element of \(X{\widehat{\otimes }}_\pi Y\) attains its projective norm. We prove that this is the case if X is the dual of a subspace of a predual of an \(\ell _1(I)\) space and Y is 1-complemented in its bidual under approximation property assumptions. This result allows us to provide some new examples where X is a Lipschitz-free space. We also prove that the set of norm-attaining elements is dense in \(X{\widehat{\otimes }}_\pi Y\) if, for instance, \(X=L_1(\mu )\) and Y is any Banach space, or if X has the metric \(\pi \) -property and Y is a dual space with the RNP.
