The Euclid ERO programme targeted the Perseus cluster of galaxies, gathering deep data in the central region of the cluster over 0.7 square degree, corresponding to approximately 0.25 r_200. The data set reaches a point-source depth of IE=28.0 (YE, JE, HE = 25.3) AB magnitudes at 5 sigma with a 0.16" and 0.48" FWHM, and a surface brightness limit of 30.1 (29.2) mag per square arcsec. The exceptional depth and spatial resolution of this wide-field multi-band data enable the simultaneous detection and characterisation of both bright and low surface brightness galaxies, along with their globular cluster systems, from the optical to the NIR. This study advances beyond previous analyses of the cluster and enables a range of scientific investigations summarised here. We derive the luminosity and stellar mass functions (LF and SMF) of the Perseus cluster in the Euclid IE band, thanks to supplementary u,g,r,i,z and Halpha data from the CFHT. We adopt a catalogue of 1100 dwarf galaxies, detailed in the corresponding ERO paper. We identify all other sources in the Euclid images and obtain accurate photometric measurements using AutoProf or AstroPhot for 138 bright cluster galaxies, and SourceExtractor for half a million compact sources. Cluster membership for the bright sample is determined by calculating photometric redshifts with Phosphoros. Our LF and SMF are the deepest recorded for the Perseus cluster, highlighting the groundbreaking capabilities of the Euclid telescope. Both the LF and SMF fit a Schechter plus Gaussian model. The LF features a dip at M(IE)=-19 and a faint-end slope of alpha_S = -1.2 to -1.3. The SMF displays a low-mass-end slope of alpha_S = -1.2 to -1.35. These observed slopes are flatter than those predicted for dark matter halos in cosmological simulations, offering significant insights for models of galaxy formation and evolution.Read less <