This paper considers a finite spectrum assignment problem for retarded non-linear systems. First, using extensions of the Lie derivative and the Lie bracket for differential difference equations, we propose a finite spectrum assignment procedure for a class of retarded non-linear systems. Then we derive a necessary and sufficient condition for the solvability of a class of finite spectrum assignment problems for retarded non-linear systems. The obtained condition is an extension of the condition for the exact linearization of finite dimensional non-linear systems and the finite spectrum assignment of retarded linear systems with controllability over polynomial rings.