Context : We have observed the late-type peculiar galaxy NGC 4424 during the Virgo Environmental Survey Tracing Galaxy Evolution ( VESTIGE ) , a blind narrow-band H \alpha + [ NII ] imaging survey of the Virgo cluster carried out with MegaCam at the Canada French Hawaii Telescope ( CFHT ) . The presence of a \sim 110 kpc long ( in projected distance ) HI tail in the southern direction indicates that this galaxy is undergoing a ram pressure stripping event . The deep narrow-band image revealed the presence of a low surface brightness ( \Sigma ( H \alpha ) \simeq 4 \times 10 ^ { -18 } erg s ^ { -1 } cm ^ { -2 } arcsec ^ { -2 } ) ionised gas tail \sim 10 kpc long extending from the centre of the galaxy to the north-west direction , thus in the direction opposite to the HI tail . Chandra and XMM X-rays data do not show any compact source in the nucleus nor the presence of an extended tail of hot gas , while IFU spectroscopy ( MUSE ) indicates that the gas is photo-ionised in the inner regions and shock-ionised in the outer parts . Medium- ( MUSE ) and high-resolution ( Fabry-Perot ) IFU spectroscopy confirms that the ionised gas is kinematically decoupled from the stellar component and indicates the presence of two kinematically distinct structures in the stellar disc . The analysis of the SED of the galaxy indicates that the activity of star formation has been totally quenched in the outer disc \sim 250-280 Myr ago , while only reduced by \sim 80 % in the central regions . All this observational evidence suggests that NGC 4424 is the remnant of an unequal-mass merger occurred \lesssim 500 Myr ago , when the galaxy was already a member of the Virgo cluster , now undergoing a ram pressure stripping event which has removed the gas and quenched the activity of star formation in the outer disc . The tail of ionised gas probably results from the outflow produced by a central starburst fed by the collapse of gas induced by the merging episode . This outflow is sufficiently powerful to overcome the ram pressure induced by the intracluster medium on the disc of the galaxy crossing the cluster . This analysis thus suggests that feedback can participate in the quenching process of galaxies in high-density regions . Aims : Methods : Results : Conclusions :