  HAT-P-67 b is one of the lowest-density gas giants known to date, making it
an excellent target for atmospheric characterization through the transmission
spectroscopy technique. In the framework of the GAPS large programme, we
collected four transit events, with the aim of studying the exoplanet
atmosphere and deriving the orbital projected obliquity. We exploited the
high-precision GIARPS observing mode of the TNG, along with additional archival
TESS photometry, to explore the activity level of the host star. We performed
transmission spectroscopy, both in the VIS and in the nIR wavelength range, and
analysed the RML effect both fitting the RVs and the Doppler shadow. Based on
the TESS photometry, we redetermined the transit parameters of HAT-P-67 b. By
modelling the RML effect, we derived a sky-projected obliquity of
($2.2\pm0.4$){\deg} indicating an aligned planetary orbit. The chromospheric
activity index $\log\,R^{\prime}_{\rm HK}$, the CCF profile, and the
variability in the transmission spectrum of the H$\alpha$ line suggest that the
host star shows signatures of stellar activity and/or pulsations. We found no
evidence of atomic or molecular species in the VIS transmission spectra, with
the exception of pseudo-signals corresponding to Cr I, Fe I, H$\alpha$, Na I,
and Ti I. In the nIR range, we found an absorption signal of the He I triplet
of 5.56$^{+0.29}_{-0.30}$%(19.0$\sigma$), corresponding to an effective
planetary radius of $\sim$3$R_p$ (where $R_p\sim$2$R_J$) which extends beyond
the planet's Roche Lobe radius. Owing to the stellar variability, together with
the high uncertainty of the model, we could not confirm the planetary origin of
the signals found in the optical transmission spectrum. On the other hand, we
confirmed previous detections of the infrared He I triplet, providing a
19.0$\sigma$ detection. Our finding indicates that the planet's atmosphere is
evaporating.
