Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\overline{\mathrm{co}}^{w^*}(C) = B_{V^*}$, where $\overline{\mathrm{co}}^{w^*}$ denotes the closed convex hull of a given set, where the closure happens in the weak-$*$ topology.

Is it possible to select a subfamily $\widetilde{C} \subseteq C$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C}) = B_{V^*}$, but, removing any element of $\widetilde{C}$ makes it so that the mentioned equality no longer holds?

I'm not sure if the usual application of the Kuratowski-Zorn's Lemma could help here, since I'm not sure if one can guarantee that the intersection of chain of subfamilies $\widetilde{C}_i$ such that $\overline{\mathrm{co}}^{w^*}(\widetilde{C_i}) = B_{V^*}$ is non-empty, or satisfies the necessary equality.