OSXDaily:

Have you received audio files or audio messages from any of your contacts on iMessage? If so, you may sometimes want to store them permanently on your iPhone or iPad so that you can listen to them later at your convenience, and save the audio attachment as a file directly. Fortunately, you can do this pretty easily, and we’ll show you how you can manually view and save audio attachments from iPhone and iPad.

As advertised, follow the sequence. Pretty low discoverability there, but not a widely used feature, I’d suspect.