Let (X, d) be a compact metric sapce and Lip(X, d) denote the Banach space of all scalar-valued Lipschitz functions f on (X, d) endowed with the norm f X,L(X,d) = max{ f X , L(X,d)( f )}, where f X = sup{| f (x)| : x ∈ X} and L(X,d)( f ) is the Lipschitz constant of f on (X, d). Applying the extreme point techniques, linear Lipschitz isometries between Lip(X, d)-spaces have been characterized in Jiménez-Vargas and Villegas-Vallecillos (J Math 34:1165–1184, 2008). In this paper we introduce norm-attaining unit functions in Lipschitz spaces and apply this consept for characterizing into and onto linear Lipschitz isometries T from Lip(X, d) to Lip(Y, ρ). In particular, we generalize the result given by Jiménez-Vargas and Villegas-Vallecillos in the surjectivity case of T by omitting the nonvanishing condition for T 1X on Y . e
