In a neutral system such as liquid helium-3, transport of mass, heat, and spin provide information analogous to electrical counterparts in metals, superconductors and topological materials. Of particular interest is transport in strongly confining channels of height approaching the superfluid coherence length, where new quantum states are found and excitations bound to surfaces and edges should be present. Here we report on the thermal conduction of helium-3 in a 1.1~μm high microfabricated channel. In the normal state we observe a diffusive thermal conductivity that is approximately temperature independent, consistent with recent work on the interference of bulk and boundary scattering. In the superfluid state we measure diffusive thermal transport in the absence of thermal counterflow. An anomalous thermal response is also detected in the superfluid which we suggest may arise from a flux of surface excitations.