The purpose of this article is to extend the Abramovich's construction of a
maximal normed extension of a normed lattice to quasi-Banach setting. It is proved that
the maximal quasi-normed extension $X^\varkappa$ of a Dedekind complete quasi-normed
lattice $X$ with the weak $\sigma$-Fatou property is a quasi-Banach lattice if and only if
$X$ is intervally complete. Moreover, $X^\varkappa$ has the Fatou and the Levi property
provided that $X$ is a Dedekind complete quasi-normed space with the Fatou property.
The possibility of applying this construction to the definition of a space of weakly
integrable functions with respect to a measure taking values from
a quasi-Banach lattice is also discussed, since the duality based definition
does not work in the quasi-Banach setting.