Atoms gain and lose electrons all the time — why never protons?

It is worth saying that proton transfer between nuclei does happen — it is simply not chemistry.

It occurs in stars, where temperatures and pressures provide the required energy and nuclei fuse. It occurs in particle accelerators, where we supply the energy deliberately. It occurs in certain kinds of radioactive decay, where a nucleus rearranges and its proton count changes, turning one element into another.

That last one is the historical punchline: transmutation of elements, which alchemists pursued for centuries and could never achieve, is real. It just requires nuclear rather than chemical processes, and the energies involved are why no arrangement of furnaces and reagents was ever going to work.

So the honest answer to your question is that it is not impossible, it is simply not available at the energies that everyday matter operates at.

Two separate reasons, and both are decisive on their own.

Energy. Removing an electron from an atom typically costs a few electron-volts. That is an amount of energy available from ordinary chemistry, from light, from heat, from friction. Removing a proton from a nucleus costs something in the region of millions of electron-volts, because you are fighting the strong nuclear force rather than electromagnetic attraction. Ordinary processes are short of the required energy by around six orders of magnitude.

Identity. Electrons are interchangeable and an atom's chemical identity does not depend on how many it has — a sodium atom that has lost one is still sodium, just charged. The number of protons is what defines the element. Add a proton to a hydrogen nucleus and you have helium, which is not a charged hydrogen atom, it is a different substance entirely.

So it is not a variation on ionisation. It would be a nuclear reaction, and it belongs to a completely different branch of physics.

There is a wrinkle worth knowing about, because it is a place where chemists really do talk about proton transfer: acid-base chemistry.

When a chemist says a proton is transferred, they usually mean a hydrogen ion — a hydrogen atom that has lost its electron and is therefore a bare proton. Moving that around is ordinary chemistry, and it is the basis of how acids and bases work.

But notice what is happening: the proton is not being extracted from a nucleus. Hydrogen's nucleus is a single proton with nothing holding it to anything else once the electron is gone, so it can move as a whole particle at chemical energies.

So the terminology collides confusingly. Chemists transfer protons constantly; none of them are removing protons from a nucleus containing more than one.

A neat way to see the scale difference: the energy released by burning a fuel and the energy released by a nuclear process differ by roughly the same factor as the two you have been comparing.

That is not a coincidence — chemistry is electrons rearranging and nuclear processes are nuclei rearranging, and the energy scales of those two are set by the forces involved. It is why a small amount of nuclear fuel corresponds to an enormous amount of chemical fuel, and it comes straight out of the answer to your question.