Based on MUSE data from the GASP survey, we study the Halpha-emitting extraplanar tails of 16 cluster galaxies at z~0.05 undergoing ram pressure stripping. We demonstrate that the dominating ionization mechanism of this gas (between 64% and 94% of the Halpha emission in the tails depending on the diagnostic diagram used) is photoionization by young massive stars due to ongoing star formation (SF) taking place in the stripped tails. This SF occurs in dynamically quite cold HII clumps with a median Halpha velocity dispersion sigma = 27 km s^-1. We study the characteristics of over 500 star-forming clumps in the tails and find median values of Halpha luminosity L_{Halpha} = 4 X 10^38 erg s^-1, dust extinction A_V=0.5 mag, star formation rate SFR=0.003 M_sun yr^-1, ionized gas density n_e =52 cm^-3, ionized gas mass M_gas = 4 X 10^4 Msun, and stellar mass M_{*} = 3 X 10^6 Msun. The tail clumps follow scaling relations (M_gas-M_{*}, L_{Halpha} -sigma, SFR-M_gas) similar to disk clumps, and their stellar masses are comparable to Ultra Compact Dwarfs and Globular Clusters.The diffuse gas component in the tails is ionized by a combination of SF and composite/LINER-like emission likely due to thermal conduction or turbulence. The stellar photoionization component of the diffuse gas can be due either to leakage of ionizing photons from the HII clumps with an average escape fraction of 18%, or lower luminosity HII regions that we cannot individually identify.
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