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 to the structure of the Banach space, as all infinite-dimensional Banach spaces have the same collection of finite-dimensional convex sets (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, for example, the paper of Fonf on Polyhedral Banach spaces.
Possibly it is worthwhile to redesign your question.