I may be wrong, but I'm not sure Apple would let something like that through, since you'd likely need permission from the other person if you wanted to record the call and there's no easy way of ensuring that you would do that. Which is why certain business and services inform you that you're being recorded before you get put through.
Yes there is. A couple of them, don't know if any are free (haven't looked into it that much) I know iSpoof would record calls. Don't know the names of any better ones of the top of my head.
Record calls with third party device Yagami, Yes, you can record calls one of many ways. If you are not looking to pay per call "Spoof," or to jailbreak your phone, you may want to consider a inline recorder. You can purchase a cheap digital recorder with input output jack for around $20-40 dollars online. This in conjunction with the earbuds that apple provides I am able to record calls. This is a quick and cheap solution the only downfall is caring around a small digital recorder. Best of luck to you and your quest to record.
Rercorder by Retronyms can record them and its .99. But you have to pay for call recording functionality