Sayyiduna Adi bin Hatim (may Allah be pleased with him) asked the Messenger of Allah (peace and blessings be upon him) about game, and he (peace and blessings be upon him) said: "If you mention the name of Allah and shoot the game with an arrow and kill it, then eat. If the game falls into water and dies, do not eat, for it is not known whether it died from the arrow or from falling into the water."
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الصيد والذبائح / 18941
Masruq (may Allah have mercy on him) narrates from Sayyiduna Abdullah (may Allah be pleased with him): When you shoot game and it falls from a mountain and dies, do not eat it, for it may have died from falling off the mountain; and if it falls into water and dies, do not eat it either, for it is possible that it was killed by the water.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الصيد والذبائح / 18942