  We provide quite sufficient conditions on the Banach spaces $E$ and $F$ in
order to obtain the spaceability of the set of all linear operators from $E$
into $F$ which are $q$-compact but not $p$-compact. Also, under similar
conditions over $E$, we prove that this set contains (up to the null operator)
a copy of $\ell_s$ whenever $F = \ell_s$. Finally, we give some applications of
our previous results to show the spaceability of some sets formed by non-linear
mappings (polynomial and Lipschitz) which are $q$-compact but not $p$-compact.
The spaceability in the space of holomorphic mappings determined by $p$-compact
sets is also considered.
