  Stellar oscillations appear all across the Hertzsprung-Russell diagram.
Recent theoretical studies support their existence also in the atmospheres of M
dwarfs. These studies predict for them short periodicities ranging from 20~min
to 3~h. Our Cool Tiny Beats (CTB) programme aims at finding these oscillations
for the very first time. With this goal, CTB explores the short time domain of
M dwarfs using radial velocity data from the HARPS-ESO and HARPS-N
high-precision spectrographs. Here we present the results for the two most
long-term stable targets observed to date with CTB, GJ~588 and GJ~699 (i.e.
Barnard's star). In the first part of this work we detail the correction of
several instrumental effects. These corrections are specially relevant when
searching for sub-night signals. Results show no significant signals in the
range where M dwarfs pulsations were predicted. However, we estimate that
stellar pulsations with amplitudes larger than $\sim0.5\,\mathrm{m\,s}^{-1}$
can be detected with a 90% completeness with our observations. This result,
along with the excess of power regions detected in the periodograms, open the
possibility of non-resolved very low amplitude pulsation signals. Next
generation more precise instrumentation would be required to detect such
oscillations. However, the possibility of detecting pulsating M-dwarf stars
with larger amplitudes is feasible due to the short size of the analysed
sample. This motivates the need for completeness of the CTB survey.
