Author’s abstract: Let \(X\) be a Banach space. The property \((\star)\) “the unit ball of \(X\) belongs to Baire(\(X\), weak)” holds whenever the unit ball of \(X^*\) is weak\(^*\)-separable; on the other hand, it is also known that the validity of \((\star)\) ensures that \(X^*\) is weak\(^*\)-separable. In this paper, we use suitable renormings of \(\ell^{\infty}({\mathbb N})\) and the Johnson–Lindenstrauss spaces to show that \((\star)\) lies strictly between the weak\(^*\)-separability of \(X^*\) and that of its unit ball. As an application, we provide a negative answer to a question raised by K.Musiał [Rend. Ist. Mat. Univ. Trieste 23, 177–262 (1991; Zbl 0798.46042).