We prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces E10 containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that SL(3,R) has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that SL(3,R) has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank groups, it is sufficient to prove strong property (T) for Sp(2,R) and its universal covering group. As consequences of our main result, it follows that for the Banach spaces X∈E10, connected higher rank simple Lie groups and their lattices have property (FX) of Bader, Furman, Gelander and Monod, and that the expanders contructed from a lattice in a connected higher rank simple Lie group do not admit a coarse embedding into X.