Is there any example of a transmission of energy in a medium that does not show wave nature?
Yes, for example heat energy is transferred through three different means:
- Radiation (which is wave like)
- Conduction (which is not wave like)
- Convection (which is not wave like)
Of course, according to the Standard Model, everything has wave-like properties, so in a way there's no escape from waves, but I believe that your question is best answered with classical phenomena.
Three basic kinds of energy transfer are usually mentioned. However I don't quite agree with Sklivvz's answer so here is my take on this issue.
I'll start with radiation because it is the most fundamental of the three and so most easily answered. It is governed by Maxwell equations which are equivalent to classical wave-equations. So all radiative effects are manifestly wave-like in nature. Nevertheless, it has to be mentioned that by using various optical materials one can obtain also dissipative solutions and lots of non-linear dispersive effects.
One could say that conduction doesn't have wave-like properties because it is governed by parabolic heat equation. However, this equation surely isn't correct because it allows for infinite speed of propagation. There exists also relativistic heat equation which admits wave-like solutions (as one would expect for any relativistic system).
As for convection, this is usually governed by some kind of fluid equation from continuum mechanics (like Navier-Stokes) and I suppose that also here one could easily obtain wave-like solution (for specific system and initial conditions).
Now to give an answer to your question: yes, there exist many situations where energy is transferred in a non-wave-like fashion (like exponential decay) but I believe it's not correct to say that any kind of above three energy transfers is or isn't wave-like in general. It depends on precise situation one is dealing with and the level of approximation one is using (like taking relativity into account).