Catalyzes the covalent attachment of the prokaryotic ubiquitin-like protein modifier Pup to the proteasomal substrate proteins, thereby targeting them for proteasomal degradation. This tagging system is termed pupylation. The ligation reaction involves the side-chain carboxylate of the C-terminal glutamate of Pup and the side-chain amino group of a substrate lysine. ATP + [prokaryotic ubiquitin-like protein]-L-glutamate + [protein]-L-lysine = ADP + phosphate + N(6)-([prokaryotic ubiquitin-like protein]-gamma-L-glutamyl)-[protein]-L-lysine. Protein degradation; proteasomal Pup-dependent pathway. Protein modification; protein pupylation. The reaction mechanism probably proceeds via the activation of Pup by phosphorylation of its C-terminal glutamate, which is then subject to nucleophilic attack by the substrate lysine, resulting in an isopeptide bond and the release of phosphate as a good leaving group. Belongs to the Pup ligase/Pup deamidase family. Pup-conjugating enzyme subfamily. Extended N-terminus.