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Single star evolution does not allow extremely low-mass stars to cross the classical instability strip (IS) during the Hubble time. However, within binary evolution framework low-mass stars can appear inside the IS once the mass transfer (MT) is taken into account. Triggered by a discovery of low-mass (0.26 M☉) RR Lyrae-like variable in a binary system, OGLE-BLGRRLYR- 02792, we investigate the occurrence of similar binary components in the IS, which set up a new class of low-mass pulsators. They are referred to as binary evolution pulsators (BEPs) to underline the interaction between components, which is crucial for substantial mass-loss prior to the IS entrance. We simulate a population of 500 000 metal-rich binaries and report that 28 143 components of binary systems experience severe MT (losing up to 90 per cent of mass), followed by at least one IS crossing in luminosity range of RR Lyrae (RRL) or Cepheid variables. A half of these systems enter the IS before the age of 4 Gyr. BEPs display a variety of physical and orbital parameters, with the most important being the BEP mass in range 0.2-0.8 M☉, and the orbital period in range 10-2 500 d. Based on the light curve only, BEPs can be misclassified as genuine classical pulsators, and as such they would contaminate genuine RRL and classical Cepheid variables at levels of 0.8 and 5 per cent, respectively. We state that the majority of BEPs will remain undetected and we discuss relevant detection limitations. [ABSTRACT FROM AUTHOR]