Let (*X*, *d*) be a compact metric sapce and $$\mathrm{Lip}(X,d)$$ denote the Banach space of all scalar-valued Lipschitz functions *f* on (*X*, *d*) endowed with the norm $$\Vert f\Vert _{X,L_{(X,d)}}=\max \{ \Vert f\Vert _X,L_{(X,d)}(f)\}$$, where $$\Vert f\Vert _X=\sup \{|f(x)|: x \in X \}$$ and $$L_{(X,d)}(f)$$ is the Lipschitz constant of *f* on (*X*, *d*). Applying the extreme point techniques, linear Lipschitz isometries between $$\mathrm{Lip}(X,d)$$-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 $$\mathrm{Lip}(X,d)$$ to $$\mathrm{Lip}(Y,\rho )$$. 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 $$ T1_{X} $$ on *Y*.