We measured gas-phase metallicity, ionisation parameter and dust extinction for 1795 representative local star-forming galaxies using integral field spectroscopy from the SDSS-IV MaNGA survey. We self-consistently derive these quantities by comparing observed line fluxes with photoionisation models using a Bayesian framework. We also present the first comprehensive study of the [SIII]$lambdalambda$9069,9532 nebular lines, which have long been predicted to be ideal tracers of the ionisation parameter. Unfortunately, we find that current photoionisation models substantially over-predict [SIII] lines intensity, while broadly reproducing other observed optical line ratios. We discuss how to nonetheless make use of the information provided by [SIII] lines by setting a prior on the ionisation parameter. Following this approach, we derive spatially-resolved maps and radial profiles of metallicity and ionisation parameter. The metallicity radial profiles are comparable with previous works, with metallicity declining toward the outer parts and a flattening in the central regions, in agreement with infall models of galaxy formation, that predict that spiral discs build up through accretion of material, which leads to an inside-out growth. On the other hand, ionisation parameter radial profiles are flat for low-mass galaxies, while their slope becomes positive as galaxy mass increases. However, the ionisation parameter maps we obtain are clumpy, especially for low-mass galaxies. Ionisation parameter is tightly correlated with the H$alpha$ equivalent width [EW(H$alpha$)], following a nearly universal relation, which we attribute to the change of the spectral shape of ionising sources due to ageing of HII regions. We derive a positive correlation between ionisation parameter and metallicity at fixed EW(H$alpha$), in disagreement with previous theoretical works expecting an anti-correlation.
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