All welding does not require melting the material. Copper to copper can be done with just pressure. And then there is forge welding. But I agree, it isn't what I normally think of as welding. From Wiki
Forge welding is a welding process of heating two or more pieces of metal and then hammering them together. The process is one of the simplest methods of joining metals and has been used since ancient times. Forge welding is versatile, being able to join a host of similar and dissimilar metals. With the invention of electrical and gas welding methods during the Industrial Revolution, forge welding has been largely replaced.
Forge welding between similar materials is caused by solid-state diffusion. This results in a weld that consists of only the welded materials without any fillers or bridging materials.
Forge welding between dissimilar materials is caused by the formation of a lower melting temperature eutectic between the materials. Due to this the weld is often stronger than the individual metals.
The temperature required to forge weld is typically 50 to 90 percent of the melting temperature. Steel welds at a lower temperature than iron. The metal may take on a glossy or wet appearance at the welding temperature. Care must be taken to avoid overheating the metal to the point that it gives off sparks from rapid oxidation (burning).