The naked-eye star 55 Cancri hosts a planetary system with five known planets, including a hot super-Earth (55 Cnc e) extremely close to its star and a farther out giant planet (55 Cnc b), found in milder irradiation conditions with respect to other known hot Jupiters. This system raises important questions on the evolution of atmospheres for close-in exoplanets, and the dependence with planetary mass and irradiation. These questions can be addressed by Lyman-α transit observations of the extended hydrogen planetary atmospheres, complemented by contemporaneous measurements of the stellar X-ray flux. In fact, planet “e” has been detected in transit, suggesting the system is seen nearly edge-on. Yet, planet “b” has not been observed in transit so far. Here, we report on Hubble Space Telescope STIS Lyα and Chandra ACIS-S X-ray observations of 55 Cnc. These simultaneous observations cover two transits of 55 Cnc e and two inferior conjunctions of 55 Cnc b. They reveal the star as a bright Lyα target and a variable X-ray source. While no significant signal is detected during the transits of 55 Cnc e, we detect a surprising Lyα absorption of 7.5 ± 1.8% (4.2σ) at inferior conjunctions of 55 Cnc b. The absorption is only detected over the range of Doppler velocities where the stellar radiation repels hydrogen atoms towards the observer. We calculate a false-alarm probability of 4.4%, which takes the a-priori unknown transit parameters into account. This result suggests the possibility that 55 Cnc b has an extended upper H i atmosphere, which undergoes partial transits when the planet grazes the stellar disc. If confirmed, it would show that planets cooler than hot Jupiters can also have extended atmospheres.