Hidden worlds: a non-transiting candidate planet in the Neptunian desert around the solar-type pulsator KIC 9139163

Close-in substellar companions experience strong tidal and magnetic interactions with their host stars and are therefore subject to fast orbital evolution. The solar-type pulsator KIC 9139163 exhibits in its light curve a stable 0.6-day modulation for which the best explanation is the presence of a close-in non-transiting companion that we therefore attempt to characterise. We combine Kepler and TESS photometric data with spectroscopic observations obtained with HARPS-N. The analysis of the radial velocities obtained with HARPS-N provides a companion mass $M_p \sin i = 7.3 \pm 1.4 \, \mathrm{M}_{\oplus}$. We infer a planetary radius of $2.43 \pm 0.14 \, \mathrm{R}_\oplus$, which, combined with the measured mass and retrieved inclination, implies a bulk density consistent with a hot water-rich world. This places the non-transiting companion candidate of KIC 9139163 within the Neptunian desert, a regime where planets are expected either to have lost their primordial hydrogen/helium envelopes or to harbour metal-enriched atmospheres. We further detect significant variations in amplitude between the Kepler and TESS phase curves, obtained six years apart, as well as a secular increase in amplitude over the Kepler baseline. Our fit favours a model with two distinct longitudinal cloud offsets over a single-offset scenario. Both datasets indicate a moderate-to-high geometric albedo and low-to-moderate heat redistribution. The opposite phase offsets observed in the Kepler and TESS datasets suggest a time-variable longitudinal brightness distribution. While making KIC 9139163 an interesting candidate for future ground follow-ups, it also suggests that searching for other non-transiting planets around fast stellar rotators in space-borne photometric surveys might provide new insights into the physics of the Neptunian planets located in the desert. [shortened]