In Brezis's Functional Analysis, there is a corollary
Corollary 3.30. Let $E$ be a separable Banach space and let $\left(f_{n}\right)$ be a bounded sequence in $E^*$. Then there exists a subsequence $\left(f_{n_{k}}\right)$ that converges in the weak topology $\sigma\left(E^*, E\right)$.
I would like to make sure if there is not any result proving the converse or a counter-example?
Let $X$ be a reflexive Banach space. As $X$ is reflexive, $X^{*}$ is also reflexive. As $X^{*}$ is reflexive, the closed unit ball of $X^{*}$ is compact in the weak topology on $X^{*}$, and consequently any scalar multiple of the closed unit ball of $X^{*}$ is compact in the weak topology on $X^{*}$.
Now take a bounded sequence in $X^{*}$. As this sequence is bounded, it is contained in a scalar multiple of the closed unit ball of $X^{*}$, which is compact in the weak topology on $X^{*}$. By the Eberlein-Šmulian theorem, the sequence has a subsequence which converges in the weak topology on $X^{*}$. As $X^{*}$ is reflexive, the weak and weak* topologies on $X^{*}$ coincide. So it follows that this sequence has a subsequence which converges in the weak* topology on $X^{*}$.
This holds for any reflexive Banach space $X$. As there exist reflexive Banach spaces which are not separable (for example, $\ell_{2}(\Gamma )$ where $\Gamma$ is an uncountable set), it follows that there are non-separable Banach spaces where every bounded sequence has a subsequence which converges in the weak* topology.
As an extra note, it is worth mentioning that if $X$ is a Banach space, you have that any bounded sequence in $X^{*}$ has a subsequence which converges in the weak* topology on $X^{*}$ if $X$ is reflexive or separable. You also have that the closed unit ball of $X^{*}$ is always compact in the weak* topology on $X^{*}$. However, it is not always the case that the closed unit ball of $X^{*}$ is sequentially compact with respect to the weak* topology on $X^{*}$. A counterexample can be obtained in the case where $X = \ell_{\infty}$. For details, see this post. So although compact and sequentially compact subsets coincide in the weak topology on a Banach space by the Eberlein-Šmulian theorem, compact and sequentially compact subsets do not coincide in the weak* topology on the dual of a Banach space in general. This also shows that there is no analogue of the Eberlein-Šmulian theorem with respect to the weak* topology.