Let me add another answer from the perspective of linear algebra/convex geometry. I think the real issue here is that the projective closed unit ball in the algebraic tensor product $E \mathbin{\otimes} F$ is, in general, not the convex hull of $B_E \mathbin{\otimes} B_F$, but rather its *closed* convex hull. If we omit the closure, even higher-dimensional faces are preserved:

**Theorem.** ([Dob20, Theorem 3.25]¹) *Let $E$ and $F$ be real vector spaces, let $C \subseteq E$, $D \subseteq F$ be symmetric convex sets, and let $M \subset C$, $N \subset D$ be proper faces. Then $\text{conv}(M \mathbin{\otimes} N)$ is a face of $\text{conv}(C \mathbin{\otimes} D)$.*

_{¹: Full disclosure: I am advertising my own paper here. The preceding theorem might have been known before, but I have not been able to find an earlier reference for this result.}

In finite-dimensional spaces, $\text{conv}(B_E \mathbin{\otimes} B_F)$ is already closed (by a compactness argument), so here it follows that the projective closed unit ball preserves proper faces. (For extreme points, this also follows from Bill Johnson's answer.)

In general, the preceding theorem does not say anything about the projective unit ball, since an extreme point of $\text{conv}(B_E \mathbin{\otimes} B_F)$ is not necessarily an extreme point of its closure $\overline{\text{conv}}(B_E \mathbin{\otimes} B_F)$.

The other answers suggest that stronger assumptions are needed to ensure preservation of extreme points. I would still be interested in a counterexample to your question, but I guess Bill Johnson's answer suffices for most practical purposes.

**References.**

[Dob20]: J. van Dobben de Bruyn, *Tensor products of convex cones, part I: mapping properties, faces, and semisimplicity*, preprint (2020), arXiv:2009.11836v1.