How to permanently save selected voicemails on iPhone?

Open the Phone App > Voicemail > Tap the message you want to save, then see the box with an up arrow and tap that. Use any of the options provided to save the message where you want it saved.

Carrier's Voicemails are stored on their server. Ask them about the process for saving it on your device or saving it permanently on their servers.

Live Voicemails are automatically saved in the Notes app. It is permanent as long as you do not delete it.