Can rummy end with rummy on the board?

No. The game is called rummy. any person who discards with rummy on the board the person who calls it gets the points. the person who is after the player who discarded the rummy then goes. it is just like floating the game is not over. when it comes back to the person who has no cards if he/she draws and can play the card they are still floating. if he/she cannot play the card that is drawn they then discard (it not being a rummy card) the game is then over.