The ionization dynamics of pure He nanodroplets irradiated by Extreme ultraviolet radiation is studied using Velocity-Map Imaging PhotoElectron-PhotoIon COincidence spectroscopy. We present photoelectron energy spectra and angular distributions measured in coincidence with the most abundant ions He(+), He2(+), and He3(+). Surprisingly, below the autoionization threshold of He droplets, we find indications for multiple excitation and subsequent ionization of the droplets by a Penning-like process. At high photon energies we observe inelastic collisions of photoelectrons with the surrounding He atoms in the droplets.