Using a compilation of H α fluxes for 384 star-forming galaxies detected during the Virgo Environmental Survey Tracing Ionised Gas Emission (VESTIGE), we study several important scaling relations linking the star formation rate, specific star formation rate, stellar mass, stellar mass surface density, and atomic gas depletion timescale for a complete sample of galaxies in a rich environment. The extraordinary sensitivity of the narrow-band imaging data allows us to sample the whole dynamic range of the H α luminosity function, from massive galaxies ( M star ≃ 10 11 M ⊙ ) to dwarf systems ( M star ≃ 10 6 M ⊙ ), where the ionised gas emission is due to the emission of single O-early B stars. This extends previous works to a dynamic range in stellar mass and star formation rate (10 −4 ≲ SFR ≲ 10 M ⊙ yr −1 ) that has never been explored so far. The main-sequence relation derived for all star-forming galaxies within one virial radius of the Virgo cluster has a slope comparable to that observed in other nearby samples of isolated objects, but its dispersion is about three times larger (∼1 dex). The dispersion is tightly connected to the available amount of HI gas, with gas-poor systems located far below objects of similar stellar mass, but with a normal HI content. When measured on unperturbed galaxies with a normal HI gas content (HI-def ≤ 0.4), the relation has a slope a = 0.92 ± 0.06, an intercept b = −1.57 ± 0.06 (at a pivot point of log M star = 8.451 M ⊙ ), and a scatter σ ≃ 0.40, and it has a constant slope in the stellar mass range 10 6 ≲ M star ≲ 3 × 10 11 M ⊙ . The specific star formation rate of HI-poor galaxies is significantly lower than that of HI-rich systems of similar stellar mass, while their atomic gas consumption timescale τ HI is fairly similar, in particular, for objects of stellar mass 10 7 ≲ M star ≲ 10 9 M ⊙ . We compare these observational results to the prediction of models expressly tuned to reproduce the effects induced by the interaction of galaxies with their surrounding environment. The observed scatter in the main-sequence relation can be reproduced only after a violent and active stripping process such as ram-pressure stripping that removes gas from the disc (outer parts first) and quenches star formation on short (< 1 Gyr) timescales. This rules out milder processes such as starvation. This interpretation is also consistent with the position of galaxies of different star formation activity and gas content within the phase-space diagram. We also show that the star-forming regions that formed in the stripped material outside perturbed galaxies are located well above the main-sequence relation drawn by unperturbed systems. These extraplanar HII regions, which might be at the origin of ultra-compact dwarf galaxies (UCDs) and other compact sources typical in rich environments, are living a starburst phase lasting only ≲50 Myr. They later become quiescent systems.