The theory of ${\mathit{sp}}^{3}$-bonded substitutional deep impurity levels is extended to type-II gap-misaligned InAs/GaSb superlattices. The theory predicts that some shallow impurities (donors or acceptors) in either bulk InAs or bulk GaSb can become deep traps in a thin-layer superlattice. This happens because the deep levels associated with point defects in either InAs or GaSb layers (when measured relative to the valence-band maximum of InAs) are much less sensitive to changes of the layer thicknesses of the superlattice than the superlattice band edges. Some common column-IV dopants may lose their amphoteric character [i.e., being a donor when substituting for the column-III host atom (${\mathit{C}}_{\mathrm{In}}$ in InAs) and an acceptor when substituting for the column-V host atom (${\mathit{C}}_{\mathrm{As}}$ in InAs)] and become deep traps for some superlattice layer thicknesses and can even become ``false valence'' dopants (e.g., ${\mathit{C}}_{\mathrm{In}}$ is predicted to be an acceptor in a 10\ifmmode\times\else\texttimes\fi{}10 InAs/GaSb [001] superlattice) for other layer thicknesses. The deep-level splitting and shifting in the type-II superlattice is found to follow the same physics as in the type-I superlattice. A semiconductor-semimetal-semiconductor transition is predicted to occur as the InAs layer thickness increases.