Existence of analytic solutions of an iterative functional differential equation is studied by reducing locally the equation to another functional differential equation without iteration of the unknown function. As done previously we first discuss the case that the constant f (0) is not on the unit circle in C and the case that the constant lies on the circle but fulfills the Diophantine condition. Furthermore, we investigate the case that the constant is a unit root, which violates the Diophantine condition.