We predict that long-lived excitons with very large binding energies can also exist in a single or few layers of monochalcogenides such as GaSe. Our theoretical study shows that excitons confined by a radial local strain field are unable to recombine despite electrons and holes coexisting in space. The localized single-particle states are calculated in the envelope function approximation based on a three-band k·p Hamiltonian obtained from density-functional-theory calculations. The binding energy and the decay rate of the exciton ground state are computed after including correlations in the basis of electron-hole pairs. The interplay between the localized strain and the caldera-type valence band characteristic of few-layered monochalcogenides creates localized electron and hole states with very different quantum numbers which hinders the recombination even for singlet excitons.