Yes, but the feet do not have to be on the ground (in soccer all "lines" effectively extend vertically into the air) or stationary. As soon as the ball is kicked the goalie can move in any direction (including forward) to stop the shot. Practically, all goalies will try to anticipate the kick and leap a split-second before the ball is struck.
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The goal keeper must stay on the goal line. Side-to-side movement is allowed, but not forward until the ball is kicked. Obviously, lifting the feet is allowed in order to move from side-to-side.