Is there any silver in a US dime?

It depends on when it was minted. Modern dimes don't contain any silver, but up till 1964 all US dimes were made of an alloy of 90% silver and 10% copper.

When the price of silver shot up in the 1960s the Mint was forced to remove silver from circulating coins, so all circulating dimes and quarters dated 1965 and later, and halves and dollars dated 1971 and later, have no silver in them.