  We construct for each $0<p\le 1$ an infinite collection of subspaces of
$\ell_p$ that extend the example from [J. Lindenstrauss, On a certain subspace
of $\ell_{1}$, Bull. Acad. Polon. Sci. S\'er. Sci. Math. Astronom. Phys. 12
(1964), 539-542] of a subspace of $\ell_{1}$ with no unconditional basis. The
structure of this new class of $p$-Banach spaces is analyzed and some
applications to the general theory of $\mathcal{L}_{p}$-spaces for $0<p<1$ are
provided. The introduction of these spaces serves the purpose to develop the
theory of conditional quasi-greedy bases in $p$-Banach spaces for $p<1$. Among
the topics we consider are the existence of infinitely many conditional
quasi-greedy bases in the spaces $\ell_{p}$ for $p\le 1$ and the careful
examination of the conditionality constants of the "natural basis" of these
spaces.
