We know from Banach-Alaoglu that if $E^*$ is separable then the unit ball is metrizable with the weak topology.
Now is this true for reflexive spaces? I.e., if a Banach $E$ (eq. $E^*$) is reflexive, is its ball weakly-metrizable?
I suspect the answer is no. Is there an interesting counter-example?
Thank you in advance.