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Stellar oscillations appear all across the Hertzsprung-Russell diagram.Recent theoretical studies support their existence also in the atmospheres of Mdwarfs. These studies predict for them short periodicities ranging from 20~minto 3~h. Our Cool Tiny Beats (CTB) programme aims at finding these oscillationsfor the very first time. With this goal, CTB explores the short time domain ofM dwarfs using radial velocity data from the HARPS-ESO and HARPS-Nhigh-precision spectrographs. Here we present the results for the two mostlong-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 ofseveral instrumental effects. These corrections are specially relevant whensearching for sub-night signals. Results show no significant signals in therange where M dwarfs pulsations were predicted. However, we estimate thatstellar 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 thepossibility of non-resolved very low amplitude pulsation signals. Nextgeneration more precise instrumentation would be required to detect suchoscillations. However, the possibility of detecting pulsating M-dwarf starswith larger amplitudes is feasible due to the short size of the analysedsample. This motivates the need for completeness of the CTB survey.

High-cadence spectroscopy of M-dwarfs - II. Searching for stellar pulsations with HARPS