  Given a bounded convex subset $C$ of a Banach space $X$ and a free
ultrafilter $\mathcal U$, we study which points $(x_i)_\mathcal U$ are extreme
points of the ultrapower $C_\mathcal U$ in $X_\mathcal U$. In general, we
obtain that when $\{x_i\}$ is made of extreme points (respectively denting
points, strongly exposed points) and they satisfy some kind of uniformity, then
$(x_i)_\mathcal U$ is an extreme point (respectively denting point, strongly
exposed point) of $C_\mathcal U$. We also show that every extreme point of
$C_{\mathcal U}$ is strongly extreme, and that every point exposed by a
functional in $(X^*)_{\mathcal U}$ is strongly exposed, provided that $\mathcal
U$ is a countably incomplete ultrafilter. Finally, we analyse the extremal
structure of $C_\mathcal U$ in the case that $C$ is a super weakly compact or
uniformly convex set.
