  [Abridged] 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. High
resolution (R$\sim$110000) time-series spectra of five transits of 55 Cancri e,
obtained with three different telescopes (UVES/VLT, HARPS/ESO 3.6m &
HARPS-N/TNG) were analysed. Targeting the sodium D lines and the calcium H and
K lines, the potential planet exospheric signal was filtered out from the much
stronger stellar and telluric signals, making use of the change of the radial
component of the orbital velocity of the planet over the transit from -57 to
+57 km/sec. Combining all five transit data sets, we detect a signal
potentially associated with sodium in the planet exosphere at a statistical
significance level of 3$\sigma$. Combining the four HARPS transits that cover
the calcium H and K lines, we also find a potential signal from ionized calcium
(4.1 $\sigma$). Interestingly, this latter signal originates from just one of
the transit measurements - with a 4.9$\sigma$ 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 <4$\sigma$) to claim unambiguous
exospheric detections. By comparing the observed signals with artificial
signals injected early in the analysis, the absorption by Na and Ca$^+$ are
estimated to be at a level of approximately 2.3$\times 10^{-3}$ and 7.0$\times
10^{-2}$ respectively, relative to the stellar spectrum.
