Interpolation of compact Lipschitz operators

Let (A_0,A_1) and (B_0,B_1) be Banach couples such that A_0 is contained in A_1 and (B_0,B_1) satisfies Arne Persson's approximation condition (H). Let T:A_1 --> B_1 be a possibly nonlinear Lipschitz mapping which also maps A_0 into B_0 and satisfies the following quantitative compactnesss condition: Ta \in ||a||_{A_0} K for each a \in A_0, where K is a fixed compact subset of B_0. We show that T maps the real interpolation space (A_0,A_1)_{\theta,p} compactly into its counterpart (B_0,B_1)_{\theta,p} for each \theta \in (0,1) and p \in [1,\infty].