Consider a neutral atom. An external force acts on one of its valence electron so that it brings this valence electron to infinity away from the rest of the atom. The electron's kinetic energy does not change during this process. In other words, at every instant of time the external force is equal (or infinitesimally equal) and opposite to the attraction force of the rest of the atom on the electron.
If we do this, the final sum of mass of the electron at infinity and the mass of the rest of the atom will be larger than the initial mass of the atom. Does this mean that at every instant of time, when the external force is applied on the electron, the mass of the electron increases? But then the net force on the electron is zero so why will its mass change?