  We show that every hyperbolic group has a proper uniformly Lipschitz affine
action on a subspace of an $L^1$ space. We also prove that every acylindrically
hyperbolic group has a uniformly Lipschitz affine action on such a space with
unbounded orbits. Our main tools are the $\mathbb{Q}$-bicombings on hyperbolic
groups constructed by Mineyev and the characterisation of acylindrical
hyperbolicity in terms of actions on quasi-trees by Balasubramanya.
