No, Amazon Music Unlimited is not free for Prime members. Amazon Music Unlimited is a paid subscription service, separate from Prime.
Prime members have access to Prime Music, a benefit of being an Amazon Prime member. Prime Music is a streaming music service, accessible through the Amazon Music app for Prime members, that has a library of over 2 million songs.
Amazon Music Unlimited is a separate subscription service, offering access to a much larger library of more than 50 million songs, ad-free and on demand with unlimited skips. It also includes more features, such as personalized stations and playlists, better audio quality and lyrics for many songs.
If you're an Amazon Prime member, you can get a discounted rate for Amazon Music Unlimited. You can get either an individual subscription for $7.99/month or $79/year, or a family membership for $14.99/month or $149/year (up to six users).
To find out more, visit the Amazon Music Unlimited page here:
https://www.amazon.com/gp/dmusic/promotions/AmazonMusicUnlimited
Good luck in finding the right streaming music service for you!