Aims. We measure far-infrared (FIR) emission from tails of stripped dust following the ionised and atomic gas components in galaxies undergoing ram pressure stripping. We study the dust-to-gas relative distribution and mass ratio in the stripped interstellar medium and relate them to those of the intra-cluster medium (ICM), thus linking the cluster-ICM-galaxy evolution at small-scales. The galaxy sample consists of three Scd Virgo galaxies with stellar masses in the range of 109 ≲ M* ≲ 1010 M⊙ and within 1 Mpc from the cluster centre, namely NGC 4330, NGC 4522, and NGC 4654. Methods. Through the analysis of Virgo Environmental Survey Tracing Ionised Gas Emission (VESTIGE) Hα, Herschel SPIRE FIR, and VLA Imaging of Virgo in Atomic gas HI data, we trace the spatial distribution of the tails and infer the dust and gas masses from the measured FIR 250 μm and HI flux densities. Dust-to-gas mass ratios in the tails are analysed as a function of the galaxy mass, metallicity, and dust temperature. Results. Along the stripped component, the dust distribution closely follows the HI and Hα emitting gas, which extend beyond the optical disc (defined by the B-band 25th magnitude isophote). In these regions, the dust-to-gas mass ratios are 2.0 ± 0.6 × 10−3, 0.7 ± 0.1 × 10−3, and 0.4 ± 0.03 × 10−3 for NGC 4330, NGC 4522, and NGC 4654, respectively. Thus, dust is widespread in the stripped material with a lower dust-to-gas mass ratio (up to a factor of 15) than the one measured in the main body of nearby galaxies. We also find a negative trend in the dust-to-gas mass ratio as a function of the metallicity that can be explained in terms of a dust component more centrally concentrated in more metal-rich systems. Together with the finding that the stripped dust is cold, Td ≲ 25 K, our results can be interpreted as a consequence of an outside-in stripping of the galaxy interstellar medium. Conclusions. Gas and dust in galaxies are perturbed in a similar fashion by the cluster environment, although their relative contribution differs from the one measured in the main body of the galaxies. When this value is considered, ram pressure stripping is consistent with being one of the key mechanisms in building up the Virgo intra-cluster component, injecting dust grains into the ICM, thus contributing to its metal enrichment.
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