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We construct analytic quasi-periodic solutions of a state-dependent delay differential equation with quasi-periodically forcing. We show that if we consider a family of problems that depends on one dimensional parameters(with some non-degeneracy conditions), there is a positive measure set Π of parameters for which the system admits analytic quasiperiodic solutions. The main difficulty to be overcome is the appearance of small divisors and this is the reason why we need to exclude parameters. Our main result is proved by a Nash-Moser fast convergent method and is formulated in the a-posteriori format of numerical analysis. That is, given an approximate solution of a functional equation which satisfies some nondegeneracy conditions, we can find a true solution close to it. This is in sharp contrast with the finite regularity theory developed in [HdlL16]. We conjecture that the exclusion of parameters is a real phenomenon and not a technical difficulty. More precisely, for generic families of perturbations, the quasi-periodic solutions are only finitely differentiable in open sets in the complement of parameters set Π.

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