Apple - Saving audio messages sent/received with iPhone

Go to Settings app > Messages and scroll down to AUDIO MESSAGES and VIDEO MESSAGES Under each one, there is an option labeled Expire. Tap on it and then tap Never to prevent them from being deleted automatically.

Never or two minutes are your only options, though.

If you want to have recording outside of iMessage (i.e. as an mp3 or whatever), then press and hold the recording in iMessage. A menu will appear with Copy, Save, More.... Select Save and your recording will now be in the Voice Memos app. From there you can tap the share button to send as email attachment, or you can get it from iTunes when you connect your phone to a computer.

In iOS 13 and above, you can select “copy” from the context menu. Then go to the Files app, tap and hold on some blank area inside one of your folders, then tap “paste”. An audio file of the message will appear. (It looks like you have to do this after tapping “keep” in the iMessage thread).