Here is a quote from the paper "Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature" by Niculescu and Rovenţa, http://www.hindawi.com/journals/fpta/2009/906727.html

[In a CAT(0) space] "the convex hull of a finite subset is not necessarily closed, but we can mention two important cases when this happens. The first one is that of Hilbert spaces. In fact, in any locally convex Hausdorff space, if are compact convex subsets, then the convex hull of their union is compact too. See the monograph of Day [9]."

Two conclusions: 1) for Hilbert spaces the result holds. 2) they claim the convex hull of a finite set is not necessarily closed, hence not necessarily compact.

Here is a quote from the paper "Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature" by Niculescu and Rovenţa, http://www.hindawi.com/journals/fpta/2009/906727.html, DOI:10.1155/2009/906727

[In a CAT(0) space] "the convex hull of a finite subset is not necessarily closed, but we can mention two important cases when this happens. The first one is that of Hilbert spaces. In fact, in any locally convex Hausdorff space, if are compact convex subsets, then the convex hull of their union is compact too. See the monograph of Day [9]."

Two conclusions: 1) for Hilbert spaces the result holds. 2) they claim the convex hull of a finite set is not necessarily closed, hence not necessarily compact.