  We prove a higher regularity result for weak solutions to nonlinear nonlocal
equations along the integrability scale of Bessel potential spaces $H^{s,p}$
under a mild continuity assumption on the kernel. By embedding, this also
yields regularity in Sobolev-Slobodeckij spaces $W^{s,p}$. Our approach is
based on a characterization of Bessel potential spaces in terms of a certain
nonlocal gradient-type operator and a perturbation approach commonly used in
the context of local elliptic equations in divergence form.
