We present a sub-kpc resolved study of the interstellar medium properties in SDP.81, a z=3.042 strongly gravitationally lensed dusty star-forming galaxy, based on high-resolution, multi-band ALMA observations of the FIR continuum, CO ladder and the [CII] line. Using a visibility-plane lens modelling code, we achieve a median source-plane resolution of ~200 pc. We use photon-dominated region (PDR) models to infer the physical conditions - far-UV field strength, density, and PDR surface temperature - of the star-forming gas on 200-pc scales, finding a FUV field strength of ~10^3-10^4 G0, gas density of ~10^5 cm^-3 and cloud surface temperatures up to 1500 K, similar to those in the Orion Trapezium region. The [CII] emission is significantly more extended than that FIR continuum: ~50 per cent of [CII] emission arises outside the FIR-bright region. The resolved [CII]/FIR ratio varies by almost 2 dex across the source, down to ~2x10^-4 in the star-forming clumps. The observed [CII]/FIR deficit trend is consistent with thermal saturation of the C+ fine-structure level occupancy at high gas temperatures. We make the source-plane reconstructions of all emission lines and continuum data publicly available.
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