\.{Z}uk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap $> \frac 1 2$, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an $L^p$-space with $1 < p < \infty$ or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap $>1-\varepsilon$. This two-sided spectral gap condition is equivalent to the fact that the Markov operator on the links has small norm. The latter is a condition that behaves well with respect to interpolation techniques, which is a key point in our arguments. Our criterion directly applies to random groups in the triangular model for densities $> \frac 1 3$, partially generalizing recent results of Drutu and Mackay. Additionally, we obtain results on the eigenvalues of $p$-Laplacians on graphs and reversible Markov chains that may be of independent interest.