The atmospheric characterisation of hot and warm Neptune-size exoplanets is challenging due to their small radius and atmospheric scale height. The warm-Neptune HAT-P-11b is a remarkable target for such characterisation due to the large brightness of its host star (V=9.46 mag; H=7.13 mag). The aims of this work are to review the main physical and architectural properties of the HAT-P-11 planetary system, and to probe the presence of 8 molecular species in the atmosphere of HAT-P-11b at high spectral resolution in the near-infrared. The planetary system was reviewed by analysing transits and occultations of HAT-P-11b from the Kepler data set as well as HIRES at Keck archival radial-velocity (RV) data. We modelled the latter with Gaussian-process regression and a combined quasi-periodic and squared-exponential kernel to account for stellar variations on both (short-term) rotation and (long-term) activity-cycle timescales. In order to probe the atmospheric composition of HAT-P-11b, we observed 4 transits of this target with GIANO-B at TNG. We find that the long-period (P~9.3 years) RV signal previously attributed to planet HAT-P-11c is more likely due to the stellar magnetic activity cycle. Nonetheless, the Hipparcos-Gaia difference in the proper-motion anomaly suggests that an outer-bound companion might still exist. For HAT-P-11b, we measure a radius Rₚ=0.4466±0.0059\,RJ, a mass Mₚ=0.0787±0.0048\,MJ, and an eccentricity e=0.2577+0.0033-0.0025, in accordance with values in the literature. Probing its atmosphere, we detect NH₃ (S/N$=5.3$, significance=5.0σ) and confirm the presence of H₂O (S/N$=5.1$, significance=3.4σ). We also tentatively detect the signal of CO₂ (S/N$=3.0$, significance=3.2σ) and CH₄ (S/N$=4.8$, significance=2.6σ), whose presence need to be confirmed by further observations.
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