James was stopping everything that Argentina could throw at him, and Luis Sacayan put a penalty over the bar to somehow keep the scores level at the half.
Dan English was replaced in the second half after taking a hand in the face, before Padilla rasped a shot just wide of the goal.
After all Argentina's dominance it could have been 1-0 at the other end but Dave Clarke just couldn't finish from James's throw.
Just before the whistle Padilla again spurned another chance, but the game ended goalless.
Britain's two draws so far mean they must get a win against Iran in their final group game in order to qualify.