  We prove that connected higher rank simple Lie groups have Lafforgue's strong
property (T) with respect to a certain class of Banach spaces
$\mathcal{E}_{10}$ containing many classical superreflexive spaces and some
non-reflexive spaces as well. This generalizes the result of Lafforgue
asserting that $\mathrm{SL}(3,\mathbb{R})$ has strong property (T) with respect
to Hilbert spaces and the more recent result of the second named author
asserting that $\mathrm{SL}(3,\mathbb{R})$ 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
$\mathrm{Sp}(2,\mathbb{R})$ and its universal covering group. As consequences
of our main result, it follows that for $X \in \mathcal{E}_{10}$, connected
higher rank simple Lie groups and their lattices have property (F$_X$) 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 $X$.
