Binding Energy and Energy Mass Equivalence 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? 
 A: The mass of the electron is the same whether it is in some bound state with an ion or free. The mass of an atom depends on its state. When an electron is gradually brought from the ground state via excited states to a continuum state the mass of the combined system increases. The potential energy increases, while the kinetic energy decreases. 
A: You are confusing classical with quantum regimes. The atom has the electrons in orbitals with definite energy, one cannot apply a continuous force of the kind you imagine to free an electron from an atom. It is interactions and not forces in the quantum regime.  Only a photon with the appropriate quantized energy will do it, (or even higher if  in a more complicated interaction of atom photon scattering).
Take a hydrogen atom. For  the electron to leave the proton at least 13.6 ev by a photon  have to be provided to the atom, and the electron will be free. From then on to get more energy it will again need interactions  with fields which can be continuous to take it to infinity by conservation of momentum, but the solutions of the hydrogen wavefunction will no longer be available to the electron.
