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:
Question. 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.
Motivation. Note that in the case of smooth manifolds, Lipschitz homotopy groups are the same as classical homotopy groups, because every homotopy can be smoothly approximated. However, for other spaces this is not the case. For example for the van Koch curve $K$ that is homeomorphic to $\mathbb{S}^1$ so we have $\pi_1(K)=\mathbb{Z}$, but Lipschitz mappings from $\mathbb{S}^1$ to $K$ are necessarily constant. Well, this example is not interesting. In order to have interesting examples we need spaces that are not smooth but have many rectifiable curves and a particularly interesting examples are given by the Heisenberg groups.
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).
