As stated your question admits an immediate answer because the extreme point structure of finite dimensional convex sets in infinite-dimensional Banach spaces is not related to the structure of the Banach space: for any such set we can find an affine (and thus, preserving extreme structure) map into any other infinite-dimensional Banach space.

Comment of Yoav Kallus is an illustration of 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.