Your question as stated does not seem to be interesting. It is clear that extreme point structure of finite dimensional convex sets in a Banach space is not related 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.