Superconducting properties change in confined geometries. Here we study the effects of strong confinement in nanosized Pb-islands on Si(111) 7×7. Small hexagonal islands with diameters less than 50 nm and a uniform height of 7 atomic layers are formed by depositing Pb at low temperature and annealing at 300 K. We measure the tunneling spectra of individual Pb-nanoislands using a low-temperature scanning tunneling microscope operated at 0.6 K, and follow the narrowing of the superconducting gap as a function of magnetic field. We find the critical magnetic field, at which the superconducting gap vanishes, reaches several Tesla, which represents a greater than 50-fold enhancement compared to the bulk value. By independently measuring the size of the superconducting gap, and the critical magnetic field that quenches superconductivity for a range of nanoislands we can correlate these two fundamental parameters and estimate the maximal achievable critical field for 7 ML Pb-nanoislands to 7 T.