  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 [BCD21]. 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 G.
Godefroy and N. Kalton. Gateaux differentiability of Lipschitz maps in the
operator space category is also studied.
