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The NASA Transiting Exoplanet Survey Satellite (TESS) is about to provide full-frame images of almost the entire sky. The amount of stellar data to be analysed represents hundreds of millions stars, which is several orders of magnitude more than the number of stars observed by the Convection, Rotation and planetary Transits satellite (CoRoT), and NASA Kepler and K2 missions. We aim at automatically classifying the newly observed stars with near real-time algorithms to better guide the subsequent detailed studies. In this paper, we present a classification algorithm built to recognise solar-like pulsators among classical pulsators. This algorithm relies on the global amount of power contained in the power spectral density (PSD), also known as the flicker in spectral power density (FliPer). Because each type of pulsating star has a characteristic background or pulsation pattern, the shape of the PSD at different frequencies can be used to characterise the type of pulsating star. The FliPer classifier (FliPerClass) uses different FliPer parameters along with the effective temperature as input parameters to feed a ML algorithm in order to automatically classify the pulsating stars observed by TESS. Using noisy TESS-simulated data from the TESS Asteroseismic Science Consortium (TASC), we classify pulsators with a 98% accuracy. Among them, solar-like pulsating stars are recognised with a 99% accuracy, which is of great interest for a further seismic analysis of these stars, which are like our Sun. Similar results are obtained when we trained our classifier and applied it to 27-day subsets of real Kepler data. FliPerClass is part of the large TASC classification pipeline developed by the TESS Data for Asteroseismology (T'DA) classification working group. [ABSTRACT FROM AUTHOR]