  We prove that cocompact (and more generally: undistorted) lattices on
$\tilde{A}_2$-buildings satisfy Lafforgue's strong property (T), thus
exhibiting the first examples that are not related to algebraic groups over
local fields. Our methods also give two further results. First, we show that
the first $\ell^p$-cohomology of an $\tilde{A}_2$-building vanishes for any
finite $p$. Second, we show that the non-commutative $L^p$-space for $p$ not in
$[\frac 4 3,4]$ and the reduced $C^*$-algebra associated to an
$\tilde{A}_2$-lattice do not have the operator space approximation property
and, consequently, that the lattice is not weakly amenable.
