The Arisen is capable of killing ghosts, and Death is actually non-corporeal. When defeated it simply banishes, but it cannot be put down forever as it is the physical manifestation of the conceptual Death in dragons dogma.
Death will flee when enough damage has been inflicted, or after a certain length of time.
Any damage inflicted on Death will carry over to every subsequent encounter until Death is slain. As the physical manifestation of the cessation of life, Death cannot be permanently slain and will eventually re-appear to resume his hunt for living prey.
In some areas, Death will disappear if the party simply exits and re-enters the dungeon. Death will also sometimes leave after slaying pawns or after a Wakestone is used."