We examine the behaviour of the infrared-radio correlation (IRRC) over the range $0<z<6$ using new, highly sensitive 3GHz observations with the Karl G. Jansky Very Large Array (VLA) and infrared data from the Herschel Space Observatory in the 2deg$^{2}$ COSMOS field. We distinguish between objects where emission is believed to arise solely from star-formation, and those where an active galactic nucleus (AGN) is thought to be present. We account for non-detections in the radio or in the infrared using a doubly-censored survival analysis. We find that the IRRC of star-forming galaxies, quantified by the infrared-to-1.4GHz radio luminosity ratio ($q_{rm TIR}$), decreases with increasing redshift: $q_{rm TIR}(z)=(2.88pm0.03)(1+z)^{-0.19pm0.01}$. Moderate-to-high radiative luminosity AGN do not follow the same $q_{rm TIR}$$(z)$ trend, having a lower normalisation and steeper decrease with redshift. We cannot rule out the possibility that unidentified AGN contributions only to the radio regime may be steepening the observed $q_{rm TIR}(z)$ trend of the star-forming population. An increasing fractional contribution to the observed 3GHz flux by free-free emission of star-forming galaxies may also affect the derived evolution. However, we find that the standard (M82-based) assumption of the typical radio spectral energy distribution (SED) for star-forming galaxies is inconsistent with our results. This suggests a more complex shape of the typical radio SED for star-forming galaxies, and that imperfect $K$ corrections in the radio may govern the derived redshift trend of $q_{rm TIR}$. Lastly, we present a redshift-dependent relation between rest-frame 1.4GHz radio luminosity and star formation rate taking the derived redshift trend into account.
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