iPhone does not have a built-in archive feature for Messages
To keep messages without deleting them, leave the conversation in the Messages app
To hide a conversation, turn on Hide Alerts for that thread
To preserve messages, make an iCloud backup of your iPhone
To save specific messages, take screenshots of the conversation
To export messages, use a Mac or third-party software
To store messages outside the iPhone, copy them into Notes, Files, or email
To prevent message loss, keep Messages in iCloud turned on
To remove a conversation from view, delete it only after saving it elsewhere
To access old messages later, search in the Messages app by contact or keyword