A counterexample is given in the following paper:

> Roberts, James W. "[A compact convex set with no extreme points][1]."
> Studia Mathematica 60.3 (1977): 255-266.

I did not see the original paper, but an exposition can be found in Section 5.6 of the book  "Metric linear spaces" by Rolewicz.


  [1]: https://eudml.org/doc/218141