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For every infinite-dimensional Banach space $X$ there is a weak*-null sequence in the unit sphere of $X^\ast$. Does this extend under suitable circumstances to the non-separable setting?

Say that $X$ is a Banach space such that the unit ball of $X^\ast$ has weak*-density $\kappa$. Is there a family $f_\alpha$ ($\alpha<\kappa)$ in the unit sphere of $X^\ast$ such that for each $x\in X$ the function $\alpha \mapsto \langle f_\alpha, x\rangle$ belongs to $c_0(\kappa)$?

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    $\begingroup$ Let me add that the property holds for the duals of reflexive spaces and more generally for the duals of $WCG$ spaces. Also remains valid for $l^{\infty}(\Gamma) $ and its dual. $\endgroup$
    – S Argyros
    Commented Feb 6 at 16:55

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A related question was asked by P. Holický, M. Šmídek, and L. Zajíček in this paper (Question 3). I have learnt from P. Koszmider and P. Hájek that the answer to this (and your) question is negative. See Example 2.16 (the space is of the form $C(K)$):

A. Dow, H. Junnila and J. Pelant, Chain conditions and weak topologies, Topology Appl. 156 (2009), 1327–1344.

For a more Banach-space theoretic exposition please consult

L. Candido, P. Koszmider, On complemented copies of $c_0(\omega_1)$ in $C(K^n)$-spaces. Stud. Math. 233 (2016), 209–226.

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