For a root of unity $ζ$ of odd prime order, we restrict coefficients of non-semisimple quantum representations of mapping class groups associated with the small quantum group $\mathfrak{u}_ζ\mathfrak{sl}_2$ from $\mathbb{Q}(ζ)$ to $\mathbb{Z}[ζ]$. We do this by exhibiting explicit bases of states spaces that span $\mathbb{Z}[ζ]$-lattices that are invariant under projective actions of mapping class groups.