Attosecond delays between photoelectron wave packets emitted from different electronic shells are now well established. Is there any delay between electrons originating from the same electronic shell but leaving the cation in different fine-structure states? This question is relevant for all attosecond photoemission studies involving heavy elements, be it atoms, molecules or solids. We answer this fundamental question by measuring energy-dependent delays between photoelectron wave packets associated with the 2 P 3 / 2 and 2 P 1 / 2 components of the electronic ground states of Xe + and Kr + . We observe delays reaching up to 33 ± 6 as in the case of Xe. Our results are compared with two state-of-the-art theories. Whereas both theories quantitatively agree with the results obtained for Kr, neither of them fully reproduces the experimental results in Xe. Performing delay measurements very close to the ionization thresholds, we compare the agreement of several analytical formulae for the continuum-continuum delays with experimental data. Our results show an important influence of spin-orbit coupling on attosecond photoionization delays, highlight the requirement for additional theory development and offer a new precision benchmark for such work.