  We prove that all 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 long-standing
conjecture of Bader, Furman, Gelander and Monod. As a consequence, we deduce
that sequences of Cayley graphs of finite quotients of a higher rank lattice
are super-expanders.
