There is an extensive literature on Lipschitz homotopies of Lipschitz maps. But I haven't seen anything about Lipschitz homotopy groups. We have introduced this notion in an article that you can find here.

Definition. Let $(X, x_0)$ be a pointed metric space. We define Lipschitz homotopy groups $\pi^{\rm Lip}_n(X, x_0)$ in the same way as classical homotopy groups, with the exception that both the maps and homotopies are required to be Lipschitz. We emphasize that we make no restriction on the Lipschitz constants. In particular, we do not require that the optimal Lipschitz constant for a homotopy between two pointed maps $f, g : ([0, 1]^n, \partial[0, 1]^n) → (X, x_0)$, be comparable to that of the maps $f$ and $g$.

This is an obvious definition, but difficult to use and perhaps this is why I could not find it anywhere. My question is: Do you know if this notion has been introduced before? I just want to be able to give credit to those who used it before.

The closest notion I can think of is the notion of Lipschitz $n$-connected spaces studied and perhaps invented by Lang and Schlichenmaier ( link to MathSciNet).