It is very easy to see that the unit ball of $c_0$ has no extreme points. I was trying to spot any extreme points in the unit ball of the space of compact operators on a Hilbert space (a *non-commutative* version of $c_0$) but without success.

Does the unit ball of $K(\ell_p)$ have any extreme points for some $p\in (1,\infty)$?