Binky is a white horse owned by Death, being a living breathing creature. Death chose him because the other horses that he tried, including the traditional skeletal horse and a fiery horse both proved impractical, as the skeletal horse kept falling apart and had to be held together by wire and the fiery horse kept setting his bedding on fire.
Binky has certain abilities associated with Death, including the ability to stand still in the air and time travel He has been used by Mortimer and Susan Sto Helit during their times as Death.
Presumably Binky will one day also die. However, because he spends much of his time in Death's Domain where time does not exist, his death is unlikely to occur for a very long time.