No, the solid rod of 1 light year length would not be able to transfer energy faster than the speed of light. According to the theory of relativity, information or signals cannot travel faster than the speed of light in a vacuum.
In your scenario, when you press the button at one end of the rod, the atoms in that part of the rod will interact with neighboring atoms, transmitting the mechanical energy through the rod. However, this propagation of energy occurs through the interactions between atoms, which are limited by the speed of sound within the material.
The speed of sound, or the propagation speed of mechanical disturbances, is much slower than the speed of light. In most solid materials, the speed of sound is typically in the range of a few kilometers per second, whereas the speed of light is approximately 299,792 kilometers per second. Therefore, the energy transmission along the rod will propagate at the speed of sound in the material, not faster than the speed of light.
Thus, pressing the button at one end of the rod would not result in the instantaneous response of the button at the other end due to the finite speed at which mechanical disturbances propagate in the material.