  We prove that for $p\ge 2$ solutions of equations modeled by the fractional
$p$-Laplacian improve their regularity on the scale of fractional Sobolev
spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$
and their gradients are in a fractional Sobolev space as well. The relevant
estimates are stable as the fractional order of differentiation $s$ reaches
$1$.
