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Building upon the linear version of mixed summable sequences in arbitraryBanach spaces of A. Pietsch, we introduce a nonlinear version of his conceptand study its properties. Extending previous work of J. D. Farmer, W. B.Johnson and J. A. Ch\'avez-Dom\'inguez, we define Lipschitz$\left(\mathfrak{m}^L\left(s;q\right),p\right)$ and Lipschitz$\left(p,\mathfrak{m}^L\left(s;q\right)\right)-$summing maps and establishinclusion theorems, composition theorems and several characterizations.Furthermore, we prove that the classes of Lipschitz$\left(r,\mathfrak{m}^L\left(r;r\right)\right)-$summing maps with $0<r<1$coincide. We obtain that every Lipschitz map is Lipschitz$\left(p,\mathfrak{m}^L\left(s;q\right)\right)-$summing map with $1\leq s< p$and $0<q\leq s$ and discuss a sufficient condition for a Lipschitz compositionformula as in the linear case of A. Pietsch. Moreover, we discuss acounterexample of the nonlinear composition formula, thus solving a problem byJ. D. Farmer and W. B. Johnson.

Lipschitz $\left(\mathfrak{m}^L\left(s;q\right),p\right)$ and $\left(p,\mathfrak{m}^L\left(s;q\right)\right)-$summing maps