For finite-dimensional $V$, your definition of a face is equivalent to the definition of a *poonem*, according to part (i) of Exercise 7 on page 21 of the book [Convex Polytopes by B. Gruenbaum][1]; then the positive answer to your question is part (iii) of Exercise 9 on the same page. 

[1]: https://archive.org/download/springer_10.1007-978-1-4613-0019-9/10.1007-978-1-4613-0019-9.pdf