I have taken the following from the review of the following paper "Schauder's conjecture on convex metric spaces" written in 2010 :
One of the most resistant open problems in the theory of nonlocally convex linear metric spaces is:
Schauder's Conjecture. Let $E$ be a compact convex subset in a topological vector space. Then any continuous mapping $f:E\to E$ has a fixed point.
In this paper, the authors prove that it holds for convex metric spaces and consequently compact convex subsets of a $CAT(0)$ space have the fixed point property for continuous mappings.
So it seems that the problem in its general form is still open.
