Tolkien book has these passages

Deer (S. Aras[1]) were graceful creatures found throughout Middle-earth. Deer were loved by the Vala Nessa, and were said to follow her as she travelled through wild lands.[2]

In the First Age the Teiglin River flowed through a gorge west of the forest of Brethil.[3] At the narrowest part of the gorge was Cabed-en-Aras, "The Deer's Leap", where once a deer leaped across the river to escape a huntsman of Haleth.[4]

In T.A. 2941[5] Thorin and Company were in the process of crossing the Enchanted River when a dark-coloured deer bowled into the Dwarves. Although Thorin shot the hart with an arrow as it leaped there was no recovering the venison for the boat they were using floated away. Worse, the commotion caused Bombur to fall into the water, which caused him to fall asleep for days after he was rescued. Soon after, a white deer and her snowy fawns appeared on the path. The dwarves shot arrows at them but none hit and the deer vanished.[6]

In Sindarin, the adjective ross was often used to describe the copper colour of red deer.[7]