Yes, with Amazon Music Unlimited or Prime, you can download songs, albums, or playlists for offline listening.
Reference : Amazon.com/code
Exploring the Features of Amazon Music
Once you have activated it, Amazon Music offers a myriad of features designed to improve the listening experience
Personalized Recommendations Explore new songs based on the music you listen to.
Stations and Playlists: Listen to curated playlists and stations that cater to different genres and moods.
Offline listening: Download songs, albums or playlists to listening offline without internet access.
High-Quality Audio Enjoy superior audio quality by using Amazon Music HD and spatial audio.
Podcasts Get access to a huge variety of audiobooks on a range of subjects.
Integration with Alexa: Control playback with voice commands from devices like your Echo devices.