We investigate first Baire functionals on the dual ball of a separable Banach space \(X\) which are pointwise limit of a sequence of \(X\) whose closed span does not contain any copy of \(\ell_1\) (or has separable dual). We propose an example of a \(C(K)\) space where not all such first Baire functionals exhibit this behavior. As an application, we study a quantitative version, in terms of descriptive set theory, of family a separable Banach spaces with this peculiarity.
