The investigation of transport properties in normal liquid helium-3 and its topological superfluid phases provides insights into related phenomena in electron fluids, topological materials, and putative topological superconductors. It relies on the measurement of mass, heat, and spin currents, due to system neutrality. Of particular interest is transport in strongly confining channels of height approaching the superfluid coherence length, to enhance the relative contribution of surface excitations, and suppress hydrodynamic counter-flow. Here we report on the thermal conduction of helium-3 in a 1.1 um high channel. The experiment was carried out by locally heating one chamber and by measuring the flow of energy out of that chamber. Figure 2) In the normal state (Figures 3, 4 and Supplemental Figures 6, 7) we observe a diffusive thermal conductivity that is approximately temperature independent, consistent with interference of bulk and boundary scattering. In the superfluid, the thermal conductivity is only weakly temperature dependent (Figure 5), requiring detailed theoretical analysis. An anomalous thermal response is detected in the superfluid (Figures 6, 7 and Supplemental Figures 2, 3, 4) which we propose arises from the emission of a flux of surface excitations from the channel. Supplemental Figure 1 summarizes calculations that show that the anomalous heat transport cannot arise from normal-superfluid counterflow. In this package we provide the data set was used to plot the figures so that digitization is not needed and the data may be used for comparison in future works.