  We extend the recent result of G. Godefroy which concerns the existence of
non-norm attaining Lipschitz maps in order to characterize the norm attainment
toward vectors for Lipschitz maps in the general setting of underlying space.
The main theorem of the present paper states that the existence of non-norm
attaining Lipschitz maps toward vectors on a large class of metric spaces is
characterized by the finite-dimensionality of range space. As an extension of
his counterexample, some denseness results on norm attaining Lipschitz maps
toward vectors are also shown.
