[en] Axion-like particles (ALPs) can induce localized oscillatory modulations in the spectra of photon sources passing through astrophysical magnetic fields. Ultra-deep Chandra observations of the Perseus cluster contain over $5\times <!--\; ??\; -->{10}^5$ counts from the active galactic nucleus (AGN) of the central cluster galaxy NGC1275 and represent a data set of extraordinary quality for ALP searches. We use this data set to search for X-ray spectral irregularities from the AGN. The absence of irregularities at the $\mathcal{O}(30\mathrm{\%<!--\; \%\; -->})$ level allows us to place leading constraints on the ALP-photon mixing parameter $g_{a\mathrm{\gamma <!--\; ??\; -->}\mathrm{\gamma <!--\; ??\; -->}}\lesssim <!--\; ???\; -->1.4\text{--}4.0\times <!--\; ??\; -->{10}^{-<!--\; ???\; -->12}{\mathrm{G}\mathrm{e}\mathrm{V}}^{-<!--\; ???\; -->1}$ for $m_a\lesssim <!--\; ???\; -->{10}^{-<!--\; ???\; -->12}\mathrm{e}\mathrm{V}$, depending on assumptions on the magnetic field realization along the line of sight.