In this paper, we investigate global uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. We shall rely on a nonlinear alternative of Leray–Schauder type in Fréchet spaces due to Frigon and Granas. The results are obtained by using the α -resolvent family ( S α ( t ) ) t ≥ 0 on a complex Banach space X combined with the above-mentioned fixed point theorem. As an application, a controllability result with one parameter is also provided to illustrate the theory.