The space $\ell_\infty(\mathcal k)$ is a dual space, and a result by Rosenthal implies that if $X^*$ contains $c_0(\Gamma)$ then it contains $\ell_\infty(\Gamma)$, and the density character of $\ell_\infty(\Gamma)$ is strictly bigger than that of $c_0(\Gamma)$.
See Corollary 1.5 in the paper H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37 (1970), 13-36. (Correction, ibid., 311-313.)