© ESO, 2014. Context. GJ 436b is one of the few transiting warm Neptunes for which a detailed characterisation of the atmosphere is possible, whereas its non-negligible orbital eccentricity calls for further investigation. Independent analyses of several individual datasets obtained with Spitzer have led to contradicting results attributed to the different techniques used to treat the instrumental effects.Aims. We aim at investigating these previous controversial results and developing our knowledge of the system based on the full Spitzer photometry dataset combined with new Doppler measurements obtained with the HARPS spectrograph. We also want to search for additional planets.Methods. We optimise aperture photometry techniques and the photometric deconvolution algorithm DECPHOT to improve the data reduction of the Spitzer photometry spanning wavelengths from 3-24 μm. Adding the high-precision HARPS radial velocity data, we undertake a Bayesian global analysis of the system considering both instrumental and stellar effects on the flux variation.Results. We present a refined radius estimate of RP = 4.10 ± 0.16 R, mass MP = 25.4 ± 2.1 M, and eccentricity e = 0.162 ± 0.004 for GJ 436b. Our measured transit depths remain constant in time and wavelength, in disagreement with the results of previous studies. In addition, we find that the post-occultation flare-like structure at 3.6 μm that led to divergent results on the occultation depth measurement is spurious. We obtain occultation depths at 3.6, 5.8, and 8.0 μm that are shallower than in previous works, in particular at 3.6 μm. However, these depths still appear consistent with a metal-rich atmosphere depleted in methane and enhanced in CO/CO2, although perhaps less than previously thought. We could not detect a significant orbital modulation in the 8 μm phase curve. We find no evidence of a potential planetary companion, stellar activity, or a stellar spin-orbit misalignment. Conclusions. Recent theoretical models invoking high-metallicity atmospheres for warm Neptunes are a reasonable match to our results, but we encourage new modelling efforts based on our revised data. Future observations covering a wide wavelength range of GJ 436b and other Neptune-class exoplanets will further illuminate their atmosphere properties, whilst future accurate radial velocity measurements might explain the eccentricity.