Buffalo
The only mammal to appear on a U.S. nickel starting in 1938 was the human being and U.S. president, Thomas Jefferson. If you're referring to the animal that appeared BEFORE 1938, that would be the famous Buffalo nickel issued from 1913 to 1938 ... but that's 25 years, not 35.
Buffalo nickels were minted between 1913 and 1938. The coin you are asking about is a Jefferson nickel with a bison reverse, part of the Westward Journey commemoratives. They are not rare and are worth face value.
minted from 1913 through 1938
Black Diamond, a famous bison from the NY Zoological Gardens, appeared on the reverse side of the United States "Indian Head" nickel from 1913 to 1938. This design was created by James Earle Fraser and showcases the majestic profile of an American buffalo.
Buffalo nickels were minted from 1913 to 1938
Buffalo nickels were minted from 1913 to 1938
5 Cents "Buffalo Nickel" 1913-1938
Buffalo nickels were minted 1913-1938.
Buffalo nickels were only struck from 1913 to 1938.
Buffalo nickels were struck from 1913 to early 1938.
That would be any Jefferson nickel dated 1938-2003 or 2006-present, as the coins feature Monticello on the reverse side (Monticello was Jefferson's house).
None of the Buffalo nickels (1913-1938) were struck in silver.