Lipschitz geometry of operator spaces and Lipschitz-free operator spaces

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