I'm wondering if a very "sparse" set in a normed vector space can look connected in the weak* topology. Specifically,

Let V be a Banach space, V* its dual, and X a (uncountable) subset of the unit ball in V* such that any two points in X are distance at least 1/2 apart from each other. Now forget the norm on V* for a moment and endow it with the weak* topology instead. Could X be connected? Could it be weak*-dense in the unit ball? Could it contain, say, the image of a closed interval under a non-constant, continuous (with respect to the weak* topology on V*, of course) map?

Interesting examples of what can or can't happen are welcome. FWIW, the space that I was looking at that prompted this question comes from bounded cohomology of groups, but finding interesting examples in, say, $L^\infty[0,1]$ would be fine by me.