Context. Neptunes represent one of the main types of exoplanets and have chemical-physical characteristics halfway between rocky and gas giant planets. Therefore, their characterization is important for understanding and constraining both the formation mechanisms and the evolution patterns of planets. Aims. We investigate the exoplanet candidate TOI-1422 b, which was discovered by the TESS space telescope around the high proper-motion G2 V star TOI-1422 (V = 10.6 mag), 155 pc away, with the primary goal of confirming its planetary nature and characterising its properties. Methods. We monitored TOI-1422 with the HARPS-N spectrograph for 1.5 yr to precisely quantify its radial velocity (RV) variation. We analyse these RV measurements jointly with TESS photometry and check for blended companions through high-spatial resolution images using the AstraLux instrument. Results. We estimate that the parent star has a radius of R_⋆ = 1.019_−0.013^+0.014 R_⊙, and a mass of M_⋆ = 1.019_−0.013^+0.014 M_⊙. Our analysis confirms the planetary nature of TOI-1422 b and also suggests the presence of a Neptune-mass planet on a more distant orbit, the candidate TOI-1422 c, which is not detected in TESS light curves. The inner planet, TOI-1422 b, orbits on a period of P_b = 12.9972 ± 0.0006 days and has an equilibrium temperature of T_eq,b = 867 ± 17 K. With a radius of R_b = 3.96_−0.11^+0.13 R_⊕, a mass of M_b = 9.0_−2.0^+2.3 M_⊕ and, consequently, a density of ρ_b = 0.795_−0.235^+0.290g cm⁻³, it can be considered a warm Neptune-sized planet. Compared to other exoplanets of a similar mass range, TOI-1422 b is among the most inflated, and we expect this planet to have an extensive gaseous envelope that surrounds a core with a mass fraction around 10% – 25% of the total mass of the planet. The outer non-transiting planet candidate, TOI-1422 c, has an orbital period of P_c = 29.29_−0.20^+0.21 days, a minimum mass, M_csin i, of 11.1_−2.3^+2.6 M_⊕, an equilibrium temperature of T_eq,c = 661 ± 13 K and, therefore, if confirmed, could be considered as another warm Neptune.