  The main result of this paper is that all affine isometric actions of higher
rank Steinberg groups over commutative rings on uniformly convex Banach spaces
have a fixed point. We consider Steinberg groups over classical root systems
and our analysis covers almost all such Steinberg groups excluding a single
rank 2 case.
  The proof of our main result stems from two independent results - a result
regarding relative fixed point properties of root subgroups of Steinberg groups
and a result regarding passing from relative fixed point properties to a
(global) fixed point property. The latter result is proven in the general
setting of groups graded by root systems and provides a far reaching
generalization of the work of Ershov, Jaikin-Zapirain and Kassabov who proved a
similar result regarding property (T) for such groups.
  As an application of our main result, we give new constructions of
super-expanders.
