We prove the hitherto hypothesized sequential dissociation of Fe(CO)$_5$ in the gas phase upon photoexcitation at 266 nm via a singlet pathway with time-resolved valence and core-level photoelectron spectroscopy with an x-ray free-electron laser. Valence photoelectron spectra are used to identify free CO molecules and to determine the time constants of stepwise dissociation to Fe(CO)$_4$ within the temporal resolution of the experiment and further to Fe(CO)$_3$ within 3 ps. Fe 3p core-level photoelectron spectra directly reflect the singlet spin state of the Fe center in Fe(CO)$_5$, Fe(CO)$_4$, and Fe(CO)$_3$ showing that the dissociation exclusively occurs along a singlet pathway without triplet-state contribution. Our results are important for assessing intra- and intermolecular relaxation processes in the photodissociation dynamics of the prototypical Fe(CO)$_5$ complex in the gas phase and in solution, and they establish time-resolved core-level photoelectron spectroscopy as a powerful tool for determining the multiplicity of transition metals in photochemical reactions of coordination complexes.