Quasi-Banach spaces of almost universal disposition

We show that for each p∈(0,1] there exists a separable p-Banach space Gp of almost universal disposition, that is, having the following extension property: for each ε>0 and each isometric embedding g:X→Y, where Y is a finite-dimensional p-Banach space and X is a subspace of Gp, there is an ε-isometry f:Y→Gp such that x=f(g(x)) for all x∈X.Such a space is unique, up to isometries, does contain an isometric copy of each separable p-Banach space and has the remarkable property of being “locally injective” amongst p-Banach spaces.We also present a nonseparable generalization which is of universal disposition for separable spaces and “separably injective”. No separably injective p-Banach space was previously known for p<1.