I will explain a recent joint work with Mikael de la Salle, in which we prove that all affine isometric actions of higher rank simple Lie groups and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This vastly generalises a recent breakthrough of Oppenheim. Combined with earlier work of Lafforgue and of Liao on strong Banach property (T) for non-Archimedean higher-rank simple groups, this confirms a conjecture of Bader, Furman, Gelander, and Monod from 2006. As a consequence, we deduce that box space expanders constructed from higher-rank lattices are superexpanders.