In this paper, we study ϕ-contractibility of natural Banach function algebras on a compact Hausdorff space. As a consequence, we characterize ϕ-contractibility of the Lipschitz algebra Lip(X, d α), for a compact metric space (X, d). We also characterize ϕ-contractibility of certain subalgebras of Lipschitz functions including rational Lipschitz algebras, analytic Lipschitz algebras and differentiable Lipschitz algebras.