We report on de Haas-van Alphen (dHvA) and band-structure studies of the iridium-pnictide superconductor BaIr₂P₂ (T_c = 2.1 K). The observed dHvA frequencies can be well understood by our band-structure calculations with two bands crossing the Fermi energy leading to a strongly corrugated Fermi-surface cylinder around the X point and a highly evolved, multiconnected Fermi surface extending over the whole Brillouin zone. The experimental effective masses are found to be considerably larger than the calculated band masses suggesting strong many-body interactions. Nevertheless, T_c remains only moderate in BaIr₂P₂ contrary to isostructural iron pnictides which probably is related to the largely different Fermi-surface topologies in these materials.