Are there any solids that undergo diffusion?
Diffusion also occurs in solids. For example, (intentional or undesirable) movement of impurity atoms or other crystal defects can be very important for the fabrication of semiconductors.

The classical experiment to demonstrate diffusion (and to measure the diffusion coefficient) in solids is the diffusion of gold in lead. A thin layer of gold that is labelled with $\ce{^198Au}$ is applied to the surface of a lead sample using electrolysis. The sample is heated, and the penetration of gold atoms into the lead sample is observed using the β radiation from $\ce{^198Au}$.

You could use a similar experiment to measure self-diffusion (e.g. of gold atoms in gold).