Absorbing an impact without recoiling?

Is it possible to absorb an impact without recoiling? For instance, would a brick wall give at all if hit by a ping-pong ball?

There will always be a recoil. In the example proposed, the wall will surely have some energy transferred to it in the form of heat and vibration. The second theorem tells us that we can only have nonzero work if the distance traveled by the point of contact is nonzero.