Context. Directly imaged planets are ideal candidates for spectroscopic characterization of their atmospheres. The angular separations that are typically close to their host stars, however, reduce the achievable contrast and thus signal-to-noise ratios (S/N). Aims. We spectroscopically characterize the atmosphere of HD 106906 b, which is a young low-mass companion near the deuterium burning limit. The wide separation from its host star of 7.1 ′′ makes it an ideal candidate for high S/N and high-resolution spectroscopy. We aim to derive new constraints on the spectral type, effective temperature, and luminosity of HD 106906 b and also to provide a high S/N template spectrum for future characterization of extrasolar planets. Methods. We obtained 1.1-2.5 µm integral field spectroscopy with the VLT/SINFONI instrument with a spectral resolution of R≈2000-4000. New estimates of the parameters of HD 106906 b are derived by analyzing spectral features, comparing the extracted spectra to spectral catalogs of other low-mass objects, and fitting with theoretical isochrones. Results. We identify several spectral absorption lines that are consistent with a low mass for HD 106906 b. We derive a new spectral type of L1.5 ± 1.0, which is one subclass earlier than previous estimates. Through comparison with other young low-mass objects, this translates to a luminosity of log(L/L ⊙ )=−3.65 ± 0.08 and an effective temperature of T eff = 1820 ± 240 K. Our new mass estimates range between M = 11.9 +1.7 −0.8 M Jup (hot start) and M = 14.0 +0.2 −0.5 M Jup (cold start). These limits take into account a possibly finite formation time, i.e., HD 106906 b is allowed to be 0-3 Myr younger than its host star. We exclude accretion onto HD 106906 b at ratesṀ > 4.8×10 −10 M Jup yr −1 based on the fact that we observe no hydrogen (Paschen-β, Brackett-γ) emission. This is indicative of little or no circumplanetary gas. With our new observations, HD 106906 b is the planetary-mass object with one of the highest S/N spectra yet. We make the spectrum available for future comparison with data from existing and next-generation (e.g., ELT and JWST) spectrographs.