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 $c_0(\kappa)$?