Your iPhone does have a voicemail password, but you only have to enter it once and many new iPhones set it up automatically. However, there still needs to be some form of authentication between your carrier and your iPhone to download your voicemails. Even though you don’t see it, your voicemail password still exists.
There is a way to save a voice message on iOS mobile by recording it on Notes or Voice Memos and set them as voicemail greeting. Also, you can share them using SMS, Email, Airdrop, Google Drive, and more, to make a back-up of the recorded voicemail. It isn't delightful when you expect a client call and can't answer it at the moment. You can't get all your phone calls at all times, especially when you are busy with your work. That is why it is important to record voicemails on your iOS device and set it as a response on an incoming call. Therefore, we have listed the steps below on how to capture voicemails on iOS.

This can easily be accomplished by setting your phone to airplane mode or turning on Do Not Disturb. Additionally, you can reject any incoming calls, sending them straight to voicemail.
While they are listening to your voice, they are internally deciding whether or not it is worth their time to continue or hang up. Look at your voicemail message as its own short advertisement.