The origin of superconductivity in the iron pnictides has been attributed to antiferromagnetic spin ordering that occurs in close combination with a structural transition, but there are also proposals that link superconductivity to orbital ordering. We used bulk-sensitive laser angle-resolved photoemission spectroscopy on BaFe(2)(As(0.65)P(0.35))(2) and Ba(0.6)K(0.4)Fe(2)As(2) to elucidate the role of orbital degrees of freedom on the electron-pairing mechanism. In strong contrast to previous studies, an orbital-independent superconducting gap magnitude was found for the hole Fermi surfaces. Our result is not expected from the superconductivity associated with spin fluctuations and nesting, but it could be better explained invoking magnetism-induced interorbital pairing, orbital fluctuations, or a combination of orbital and spin fluctuations. Regardless of the interpretation, our results impose severe constraints on theories of iron pnictides.