Fourier theory of thermal transport considers heat transport as a diffusive process where energy flow is driven by a temperature gradient. However, this is not valid at length scales smaller than the mean free path for the energy carriers in a material, which can be hundreds of nanometres in crystalline materials at room temperature. In this case, heat flow will become 'ballistic'--driven by direct point-to-point transport of energy quanta. Past experiments have demonstrated size-dependent ballistic thermal transport through nanostructures such as thin films, superlattices, nanowires and carbon nanotubes. The Fourier law should also break down in the case of heat dissipation from a nanoscale heat source into the bulk. However, despite considerable theoretical discussion and direct application to thermal management in nanoelectronics, nano-enabled energy systems and nanomedicine, this non-Fourier heat dissipation has not been experimentally observed so far. Here, we report the first observation and quantitative measurements of this transition from diffusive to ballistic thermal transport from a nanoscale hotspot, finding a significant (as much as three times) decrease in energy transport away from the nanoscale heat source compared with Fourier-law predictions.