An affine and isometric action \(\alpha\) of a countable discrete group \(\Gamma\) on a Banach space \(X\) is said to be proper if \(\lim_{g\to\infty}\| \alpha(g)\xi\| =\infty\) for every \(\xi\in X\). The purpose of this paper is to prove the following result. Theorem. If \(\Gamma\) is a hyperbolic group [cf. M.Gromov, in: Essays in group theory, Publ., Math. Sci. Res. Inst. (MSRI) 8, 75–263 (1987; Zbl 0634.20015)], then there exists a constant \(p\) depending on \(\Gamma\), \(2\leq p <\infty\), such that \(\Gamma\) admits a proper affine isometric action on an \(\ell^p\)-space. Moreover, \(p\) is strictly greater than 2 if the hyperbolic \(\Gamma\) is infinite and has property (T).