I have taken the following from the review of the following paper "[Schuader's conjecture on convex metric spaces](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=ALLF&pg7=ALLF&pg8=ET&review_format=html&s4=&s5=Schauder%20conjecture&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=2778675)" 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.