Thermal conductivity of carbon nanotube (CNT) films and micro-fibres synthesised by floating catalyst chemical vapour deposition was measured by the parallel thermal conductance method. CNT films showed in-plane thermal conductivities of 110 W m−1 K−1. Online condensation into a micro-fibre morphology – a two-dimensional reduction in the transverse plane, including some axial stretching during solvent evaporation – resulted in room-temperature thermal conductivity values as high as 770 ± 10 W m−1 K−1, which is the highest thermal conductivity reported for CNT bulk materials to date. In specific terms, this matches the maximum thermal conductivity of heat-treated carbon fibre, but with a higher onset temperature for Umklapp scattering processes (300 K rather than 150 K). We selected four sample types to investigate effects of alignment, purity, and single- or multi-wall character on their thermal conductivity. For both the electrical and thermal conductivity of as-spun material, i.e. without any post-synthesis treatment, we show that the density and quality of CNT bundle alignment are still the predominant factors affecting these properties, outweighing the influence of single- or multi-walled character of the nanotubes. This raises the promise that, with optimal alignment and junction points, even higher values of thermal conductivity are achievable for macroscopic CNT fibres.