Instagram does not allow you to back up chats, images, and other content.
To back up Instagram chats on the web: sign in, go to Your Activity, select Download your information, choose Messages, and download the file.
To back up Instagram chats on mobile: go to your profile page, open the Hamburger menu, tap Accounts Center, select Download your information, choose Messages, and download the file.
Follow the provided steps for desktop and mobile to back up Instagram chats and prevent data loss.