  We construct affine uniformly Lipschitz actions on $\ell^1$ and $L^1$ for
certain groups with hyperbolic features. For acylindrically hyperbolic groups,
our actions have unbounded orbits, while for residually finite hyperbolic
groups and for mapping class groups, the actions have proper orbits, with the
induced $L^1$-metric quasi-isometric (respectively, almost quasi-isometric) to
the word metric.
