We report the discovery of robust spectroscopically confirmed Balmer break (BB) galaxies and candidates, with secure spectroscopic redshifts $7.1 z 9.6$, from publicly available James Webb Space Telescope (JWST) extra-galactic photometric and spectroscopic surveys. To achieve this, we used dedicated filters probing the BB and inspected the objects with NIRSpec spectroscopy. We have recovered the previously known objects with strong BBs and here reveal 10-11 new objects with clear BBs, thus tripling the number of spectroscopically confirmed galaxies with a BB at $z >7$. Approximately half of them show a pure BB and no signs of recent star formation, whereas the other half show BB and emission lines, most likely indicating galaxies whose star formation ceased earlier and has restarted recently. Overall, we find that $ 10-20$<!PCT!> of all galaxies from our sample show signatures of an evolved stellar population. Furthermore, we find that the strength of the BB does not significantly depend on the rest-UV and rest-optical brightness of these sources. In short, our work confirms that photometry alone has the potential to measure BB strengths and to identify evolved stellar populations at high redshift, and that such objects may be more frequent than previously thought. The presence of galaxies with a range of break strengths and the joint presence of BB and emission lines indicate a bursty nature of the star formation in the early Universe.