  For $0<p<1,$ we prove that there is a $\mathfrak{c}$-dimensional subspace of
$\mathcal{L}\left( \ell_{p},\ell_{p}\right) $ such that, except for the null
vector, all of its vectors fail to be absolutely $(r,s)$-summing regardless of
the real numbers $r,s$, with $1\leq s\leq r<\infty$. This extends a result
proved by Maddox in 1987. Moreover, the result is sharp in the sense that it is
not valid for $p\geq1.$
