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We prove that cocompact (and more generally: undistorted) lattices on$\tilde{A}_2$-buildings satisfy Lafforgue's strong property (T), thusexhibiting the first examples that are not related to algebraic groups overlocal fields. Our methods also give two further results. First, we show thatthe first $\ell^p$-cohomology of an $\tilde{A}_2$-building vanishes for anyfinite $p$. 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 propertyand, consequently, that the lattice is not weakly amenable.

Strong Property (T), weak amenability and $\ell^p$-cohomology in $\tilde{A}_2$-buildings