Now, back to those special memories. If you find yourself listening to an old voicemail over and over — for sentimental reasons or otherwise — don't bet on your phone to save it for you indefinitely.
Unfortunately, there’s no built-in button in your iPhone’s phone app for recording calls. Instead, you’ll have to use a third-party app or service or use a clever workaround.

Forget about messing around with apps. There’s a powerful voicemail handling system built into nearly every big carrier today — and that system almost always includes a system for recording a voice message and sending it directly without ringing the target phone. I will walk you through the process of doing this on Straight Talk (since that happens to be my carrier) but most carriers will have similar systems.
5 Best Methods to Record Phone Calls on iPhone. Using Voicemail. Using Google Voice. Using a Third-party call recorder. Using the speakerphone function. Using an external recording device. Method 1. Using Voicemail. Using voicemail is the simplest way to record phone calls on iPhone without installing any app.