\.{Z}uk proved that if a finitely generated group admits Cayley graph such the Laplacian on links of this has spectral gap $> \frac{1}{2}$, then property (T), or equivalently, every affine isometric action Hilbert space fixed point. We prove same holds for actions uniformly curved Banach (for example an $L^p$-space with $1 < p \infty$ interpolation between and arbitrary space) as soon two-sided...