This article provides instructions on how to save text messages on an iPhone running iOS 14 or later. You'll learn how to save individual messages and entire conversations, among other things.
How to Save a Text Message Conversation on iPhone
Strictly speaking, there is no way to select a message or group of messages and then save them on the iPhone. You can save text messages, you just have to use a workaround.
One way to save messages on iPhone is to take screenshots of them. This method is useful if you want to save the timestamps and layout of the messages.
To take screenshots of your messages, open the thread you want to capture. Then, press the Volume Up and Home buttons on your phone at the same time. This will take a screenshot of the messages displayed on the screen.