A very well known theorem states that if $X$ is a separable banach space, $(B_{X'},\sigma(X',X))$ is metrizable and thus both second countable and separable. Is there any example of a banach space $X$ with $d((B_{X'},\sigma(X',X)))=\aleph_0<w((B_{X'},\sigma(X',X)))$ (where $w$ stands for the topological weight of the space)? Can anyone point me to a reference?
I only know this cannot happen if $X$ is reflexive, since in that case $(B_{X'},\sigma(X',X))$ is an Eberlein compact and thus strongly monolithic.