If the figure skater pushes off the wall of the imaginary frictionless rink and does not move her body (meaning she keeps her position relative to her own center of mass), she will start to move in the opposite direction due to Newton's third law of motion, which states that for every action, there is an equal and opposite reaction.
When she pushes off the wall, she exerts a force on the wall, and the wall exerts an equal and opposite force on her. This push will cause her to accelerate away from the wall. If there are no other external forces acting on her (like friction, air resistance, etc.), she will continue to move in that direction indefinitely, as per the principle of inertia. In a frictionless environment, she will maintain a constant velocity after the initial push until an external force acts on her.