Lipschitz geometry of operator spaces and Lipschitz-free operator spaces

We show that there is an operator space notion of Lipschitz embeddability between operator spaces which is strictly weaker than its linear counterpart but which is still strong enough to impose linear restrictions on operator space structures. This shows that there is a nontrivial theory of nonlinear geometry for operator spaces and it answers a question in Braga et al. (Proc Am Math Soc 149(3):1139-1149, 2021). For that, we introduce the operator space version of Lipschitz-free Banach spaces and prove several properties of it. In particular, we show that separable operator spaces satisfy a sort of isometric Lipschitz-lifting property in the sense of Godefroy and Kalton. Gateaux differentiability of Lipschitz maps in the operator space category is also studied.