  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
$0<p\le1$, denoted $\mathcal F_{p}(\mathcal M)$. Here we develop new techniques
to show that, by analogy with the case $p=1$, the space $\ell_{p}$ 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 $\ell_{p}$ 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 $\mathcal
F_{p}([0,1])$, we obtain the first-known examples of $p$-Banach spaces for
$p<1$ that possess a basis but fail to have an unconditional basis.
