This Letter presents a search for highly ionizing magnetic monopoles in 262 μb^{-1} of ultraperipheral Pb+Pb collision data at sqrt[s_{NN}]=5.36 TeV collected by the ATLAS detector at the LHC. A new methodology that exploits the properties of clusters of hits reconstructed in the innermost silicon detector layers is introduced to study highly ionizing particles in heavy-ion data. No significant excess above the background, which is estimated using a data-driven technique, is observed. Using …