Let (X, d) be a pointed compact metric space with the base point x0 and let Lip((X, d), x0) (lip((X, d), x0)) denote the pointed (little) Lipschitz space on (X, d). In this paper, we prove that every weakly compact composition operator uCφ on Lip((X, d), x0) is compact provided that lip((X, d), x0) has the uniform separation property, φ is a base point preserving Lipschitz self-map of X and u ∈ Lip(X, d) with u(x) ̸= 0 for all x ∈ X\{x0}.d
