We have obtained deep optical, long‐slit spectrophotometry of the Galactic H ii regions M 17, NGC 3576 and of the Magellanic Cloud H ii regions 30 Doradus, LMC N11B and SMC N66, recording the optical recombination lines (ORLs) of C ii, N ii and O ii. A spatial analysis of 30 Doradus is performed, revealing that the forbidden‐line [O iii] electron temperature is remarkably constant across the nebula. The forbidden‐line O2+/H+ abundance mapped by the [O iii]λ4959 collisionally excited line (CEL) is shown to be consistently lower than the recombination‐line abundance mapped by the O ii V1 multiplet at 4650 Å. In addition, the spatial profile of the C2+/O2+ ratio derived purely from recombination lines is presented for the first time for an extragalactic nebula. Temperature‐insensitive ORL C2+/O2+ and N2+/O2+ ratios are obtained for all nebulae except SMC N66. The ORL C2+/O2+ ratios show remarkable agreement within each galactic system, while also being in agreement with the corresponding CEL ratios. The disagreement found between the ORL and CEL N2+/O2+ ratios for M 17 and NGC 3576 can be attributed to the N ii V3 and V5 ORLs that were used being affected by fluorescent excitation effects. For all five nebulae, the O2+/H+ abundance derived from multiple O ii ORLs is found to be higher than the corresponding value derived from the strong [O iii]λλ4959, 5007 CELs, by factors of 1.8 to 2.7 for four of the nebulae. The LMC N11B nebula exhibits a more extreme discrepancy factor for the O2+ ion, ∼5. Thus, these H ii regions exhibit ORL/CEL abundance discrepancy factors that are similar to those previously encountered amongst planetary nebulae. Our optical CEL O2+/H+ abundances agree to within 20–30 per cent with published O2+/H+ abundances that have been obtained from observations of infrared fine‐structure lines. Since the low excitation energies of the latter make them insensitive to variations about typical nebular temperatures, fluctuations in temperature are ruled out as the cause of the observed ORL/CEL O2+ abundance discrepancies. We present evidence that the observed O ii ORLs from these H ii regions originate from gas of very similar density (<3500 cm−3) to that emitting the observed heavy‐element optical and infrared CELs, ruling out models that employ high‐density ionized inclusions in order to explain the abundance discrepancy. We consider a scenario whereby much of the heavy‐element ORL emission originates from cold (≤500 K) metal‐rich ionized regions. These might constitute haloes that are being evaporated from much denser neutral cores. The origin of these metal‐rich inclusions is not clear – they may have been ejected into the nebula by evolved, massive Of and Wolf–Rayet stars, although the agreement found between heavy‐element ion ratios derived from ORLs with the ratios derived from CELs provides no evidence for nuclear‐processed material in the ORL‐emitting regions.