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 $5\times 105$ 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 $O(30\%)$ level allows us to place leading constraints on the ALP-photon mixing parameter $ga\gamma \gamma \lesssim 1.4--4.0\times 10-12\mathrm{GeV}-1$ for $ma\lesssim 10-12\mathrm{eV}$, depending on assumptions on the magnetic field realization along the line of sight.