There are such examples already in Riemannian world! In fact in any generic Riemannian manifold of dimension $\ge3$ convex hull of 3 points in general position is not closed. BUT it is hard to make explicit and generic at the same time :)
If it is closed then there are a lot of geodesics lying in its boundary --- that is rare! To see it do the following exercise first: Show that in generic 3-dimensional manifold, arbitrary smooth convex surface contains no geodesic. (Here geodesic = geodesic in ambient space.)
To make word "generic" more clear: show that any metric admits $C^\infty$-perturbation such that above property holds.
Hint: Use Jacobi fields to show the following: If geodesic $\gamma$ lies in a convex surface then curvature tensor along $\gamma$ is very special.
P.S. Roughly it means that convex hulls in Riemannian world are too complicated. But I know one example where it is used, see Kleiner's An isoperimetric comparison theorem. But he is only using that Gauss curvature of non-extremal points on the boundary of convex hulls is zero...
Appendix. (A construction of convex hull.) To construct convex hull you can do the following: start with some set $K_0$ and construct a sequence of sets $K_n$ so that $K_{n+1}$ is a union of all geodesics with ends in $K_n$. The union $W$ of all $K_n$ is convex hull. Now assume it coincides with its closure $\bar w$. In particular if $x\in\partial\bar W$ then $x\in K_n$ for some $n$. I.e. there is a geodesic in $\bar W$ passing through $x$ (if $x\not\in K_0$). From convexity, it is clear that such geodesic lies in $\partial \bar W$...