Despite the thousands of planets in orbit around stars known to date, the mechanisms of planetary formation, migration, and atmospheric loss remain unresolved. In this work, we confirm the planetary nature of a young Saturn-size planet transiting a solar-type star every 8.03 d, TOI-1135 b. The age of the parent star is estimated to be in the interval of 125-1000 Myr based on various activity and age indicators, including its stellar rotation period of 5.13 ± 0.27 days and the intensity of photospheric lithium. We obtained follow-up photometry and spectroscopy, including precise radial velocity measurements using the CARMENES spectrograph, which together with the TESS data allowed us to fully characterise the parent star and its planet. As expected for its youth, the star is rather active and shows strong photometric and spectroscopic variability correlating with its rotation period. We modelled the stellar variability using Gaussian process regression. We measured the planetary radius at 9.02 ± 0.23 R⊕ (0.81 ± 0.02 RJup) and determined a 3σ upper limit of < 51.4 M⊕ (< 0.16 MJup) on the planetary mass by adopting a circular orbit. Our results indicate that TOI-1135 b is an inflated planet less massive than Saturn or Jupiter but with a similar radius, which could be in the process of losing its atmosphere by photoevaporation. This new young planet occupies a region of the mass-radius diagram where older planets are scarse, and it could be very helpful to understanding the lower frequency of planets with sizes between Neptune and Saturn.