Aims. We aim at detecting water vapor in the atmosphere of the hot Jupiter HD 209458 b and perform a multi-band study in the near infrared with CARMENES. Methods. The water vapor absorption lines from the atmosphere of the planet are Doppler-shifted due to the large change in its radial velocity during transit. This shift is of the order of tens of km s⁻¹, whilst the Earth’s telluric and the stellar lines can be considered quasi-static. We took advantage of this shift to remove the telluric and stellar lines using SYSREM, which performs a principal component analysis including proper error propagation. The residual spectra contain the signal from thousands of planetary molecular lines well below the noise level. We retrieve the information from those lines by cross-correlating the residual spectra with models of the atmospheric absorption of the planet. Results. We find a cross-correlation signal with a signal-to-noise ratio (S/N) of 6.4, revealing H₂O in HD 209458 b. We obtain a net blueshift of the signal of –5.2 _−1.3^+2.6 km s⁻¹ that, despite the large error bars, is a firm indication of day- to night-side winds at the terminator of this hot Jupiter. Additionally, we performed a multi-band study for the detection of H₂O individually from the three near infrared bands covered by CARMENES. We detect H₂O from its 0.96–1.06 μm band with a S/N of 5.8, and also find hints of a detection from the 1.06–1.26 μm band, with a low S/N of 2.8. No clear planetary signal is found from the 1.26–1.62 μm band. Conclusions. Our significant H₂O signal at 0.96–1.06 μm in HD 209458 b represents the first detection of H₂O from this band individually, the bluest one to date. The unfavorable observational conditions might be the reason for the inconclusive detection from the stronger 1.15 and 1.4 μm bands. H₂O is detected from the 0.96–1.06 μm band in HD 209458 b, but hardly in HD 189733 b, which supports a stronger aerosol extinction in the latter, in line with previous studies. Future data gathered at more stable conditions and with larger S/N at both optical and near-infrared wavelengths could help to characterize the presence of aerosols in HD 209458 b and other planets.