The President’s current term of office ends on January 20, 2021 no matter what. Under no circumstances may the President’s current term be extended. An incumbent president may remain in office after the expiration of his first term only by winning re-election for a second term.
In every previous presidential election, the winner was determined by the time the prior term expired. But the Constitution has a mechanism to deal with a situation where that’s not true. If, for whatever reason, the process for selecting the President has not concluded by the beginning of the next term, then the Twentieth Amendment provides that the Vice President shall “act as President” until a new President is chosen. If neither the President or the Vice President has been chosen by January 20–which could happen because the presidential electors vote for a ticket of a President and a Vice President together–then the next in line to “act as President” is the Speaker of the House.