Embeddability of ${\mathrm{\ell }}_p$ and bases in Lipschitz free $p$-spaces for $0<p\le 1$

Our goal in this paper is to continue the study initiated by the authors in [Lipschitz free $p$-spaces for $0<p<1$; arXiv:1811.01265 [math.FA]] of the geometry of the Lipschitz free $p$-spaces over quasimetric spaces for $0<p\le 1$, denoted ${\mathcal{F}}_p(\mathcal{M})$. Here we develop new techniques to show that, by analogy with the case $p=1$, the space ${\mathrm{\ell }}_p$ embeds isomorphically in ${\mathcal{F}}_p(\mathcal{M})$ for $0<p<1$. Going further we see that despite the fact that, unlike the case $p=1$, this embedding need not be complemented in general, complementability of ${\mathrm{\ell }}_p$ in a Lipschitz free $p$-space can still be attained by imposing certain natural restrictions to $\mathcal{M}$. As a by-product of our discussion on basis in $\mathcal F_{p}([0,1])$,weobtainthefirst-knownexamplesof$p$-Banachspacesfor$p<1$ that possess a basis but fail to have an unconditional basis.