Producing positronium (Ps) in the metastable $2^3text{S}$ state is of interest for various applications in fundamental physics. We report here about an experiment in which Ps atoms are produced in this long-lived state by spontaneous radiative decay of Ps excited to the $3^3text{P}$ level manifold. The Ps cloud excitation is obtained with a UV laser pulse in an experimental vacuum chamber in presence of guiding magnetic field of 25 mT and an average electric field of 300 V/cm. The indication of the $2^3text{S}$ state production is obtained from a novel analysis technique of single-shot positronium annihilation lifetime spectra. Its production efficiency relative to the total amount of formed Ps is evaluated by fitting a simple rate equations model to the experimental data and found to be $ (2.1 pm 1.3) , % $.