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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 thegeometry of the Lipschitz free $p$-spaces over quasimetric spaces for$0<p\le1$, denoted $\mathcal F_{p}(\mathcal M)$. Here we develop new techniquesto show that, by analogy with the case $p=1$, the space $\ell_{p}$ embedsisomorphically in $\mathcal F_{p}(\mathcal M)$ for $0<p<1$. Going further wesee that despite the fact that, unlike the case $p=1$, this embedding need notbe complemented in general, complementability of $\ell_{p}$ 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 $\mathcalF_{p}([0,1])$, we obtain the first-known examples of $p$-Banach spaces for$p<1$ that possess a basis but fail to have an unconditional basis.

Embeddability of $\ell_{p}$ and bases in Lipschitz free $p$-spaces for $0