Does this argument work? Let $X$ be a normed vector space, with $A$ weakly compact in $X$. The collection of sets of the form
$\{x \in X: |f(x) - f(a)| < 1 \},f \in X^*,a \in A$
forms a cover of $A$ consisting of weakly open sets in $X$, and so should has a finite subcover.