Your question as stated lacks contents. It is clear that extreme point structure of finite dimensional convex sets in a Banach space has nothing to do with the structure of the Banach space, and the example mentioned in the comment illustrates this. On the other hand, there is a very interesting theory of extreme points of unit balls of Banach spaces, in which $c_0$ plays an important role. See the paper of Fonf on Polyhedral Banach spaces.

I would suggest that you either delete or redesign your question.