The VISTA near-infrared YJKs survey of the Magellanic System (VMC) is collecting deep Ks-band time-series photometry of pulsating stars hosted by the two Magellanic Clouds and their connecting bridge. Here, we present Y, J, Ks light curves for a sample of 717 Small Magellanic Cloud (SMC) Classical Cepheids (CCs). These data, complemented with our previous results and V magnitude from literature, allowed us to construct a variety of period-luminosity and period-Wesenheit relationships, valid for Fundamental, First and Second Overtone pulsators. These relations provide accurate individual distances to CCs in the SMC over an area of more than 40 deg^2^. Adopting literature relations, we estimated ages and metallicities for the majority of the investigated pulsators, finding that (i) the age distribution is bimodal, with two peaks at 120+/-10 and 220+/-10Myr; (i) the more metal-rich CCs appear to be located closer to the centre of the galaxy. Our results show that the three-dimensional distribution of the CCs in the SMC is not planar but heavily elongated for more than 25-30kpc approximately in the east/north-east towards south-west direction. The young and old CCs in the SMC show a different geometric distribution. Our data support the current theoretical scenario predicting a close encounter or a direct collision between the Clouds some 200Myr ago and confirm the presence of a Counter-Bridge predicted by some models. The high-precision three-dimensional distribution of young stars presented in this paper provides a new test bed for future models exploring the formation and evolution of the Magellanic System.