Not sure whether this counts as recent enough:

Robert Cauty proved 2001 the Schauder conjecture that every continuous map of a nonempty commpact convex subset of a topological vector space (not necessarily locally convex!) has a fixed point:

- Cauty, Robert, *Solution du problème de point fixe de Schauder*, Fundamenta Mathematica 170, 2001, 231-246.

Although some problems have been found in the original paper, it seems that they could all be fixed.