Under CH there exists an example of a non-metrizable compact scattered Hausdorff space $K$ such that the Banach space $X=C(K)$ endowed with the weak topology is hereditarily Lindelof. The non-metrizability of $K$ implies that the Banach space $X=C(K)$ is not separable and then the dual $X^*$ is not separable as well. This example is due to Kunen and is described in the survey paper of Negrepontis in "Handbook of Set-Theoretic Topology" (1984).