Does the soap become ritually impure by washing hands with it after exiting the lavatory, and if it does become impure, how is it purified?

The fundamental principle is that the hand does not become impure after using water, and the lingering smell on the hand does not render the soap impure. Therefore, one should not open the door to obsessive doubts (waswasa). However, if the very substance of impurity (ayn al-najāsa) touches the soap, then it becomes impure. It can be purified by washing it with water and removing the substance of the impurity and its trace.