Context. The atmospheric and surface characterization of rocky planets is a key goal of exoplanet science. Unfortunately, the measurements required for this are generally out of reach of present-day instrumentation. However, the planet Mercury in our own solar system exhibits a large exosphere composed of atomic species that have been ejected from the planetary surface by the process of sputtering. Since the hottest rocky exoplanets known so far are more than an order of magnitude closer to their parent star than Mercury is to the Sun, the sputtering process and the resulting exospheres could be orders of magnitude larger and potentially detectable using transmission spectroscopy, indirectly probing their surface compositions. Aims. The aim of this work is to search for an absorption signal from exospheric sodium (Na) and singly ionized calcium (Ca⁺) in the optical transmission spectrum of the hot rocky super-Earth 55 Cancri e. Although the current best-fitting models to the planet mass and radius require a possible atmospheric component, uncertainties in the radius exist, making it possible that 55 Cancri e could be a hot rocky planet without an atmosphere. Methods. High resolution (R ~ 110 000) time-series spectra of five transits of 55 Cancri e, obtained with three different telescopes (UVES/VLT, HARPS/ESO 3.6 m and HARPS-N/TNG) were analysed. Targeting the sodium D lines and the calcium H and K lines, the potential planet exospheric signal was filtered out from the much stronger stellar and telluric signals, making use of the change of the radial component of the orbital velocity of the planet over the transit from -57 to +57 km s^-1. Results. Combining all five transit data sets, we detect a signal potentially associated with sodium in the planet exosphere at a statistical significance level of 3σ. Combining the four HARPS transits that cover the calcium H and K lines, we also find a potential signal from ionized calcium (4.1σ). Interestingly, this latter signal originates from just one of the transit measurements - with a 4.9σ detection at this epoch. Unfortunately, due to the low significance of the measured sodium signal and the potentially variable Ca⁺ signal, we estimate the p-values of these signals to be too high (corresponding to + are estimated to be at a level of ~2.3 × 10^-3 and ~7.0 × 10^-2 respectively, relative to the stellar spectrum. Conclusions. If confirmed, the 3σ signal would correspond to an optically thick sodium exosphere with a radius of 5 _R, which is comparable to the Roche lobe radius of the planet. The 4.9σ detection of Ca⁺ in a single HARPS data set would correspond to an optically thick Ca⁺ exosphere approximately five times larger than the Roche lobe radius. If this were a real detection, it would imply that the exosphere exhibits extreme variability. Although no formal detection has been made, we advocate that probing the exospheres of hot super-Earths in this way has great potential, also knowing that Mercury's exosphere varies significantly over time. It may be a fast route towards the first characterization of the surface properties of this enigmatic class of planets.