How to archive offline in Mail

You can create a folder in apple mail application in iCloud section , and from your inbox select a mail move to that folder .

If you take iCloud as offline and move the message in offline mode after a while it automatically gets converted to online ( the bold icon disappears ) .

And if you iCloud storage is full see this article Manage your iCloud storage - Apple Support

In the Mail program, go to Mail/Mailbox/New Mailbox and create a new On My Mac mailbox. Drag the emails from your iCloud mailbox to the new mailbox. If necessary, go back to iCloud mailbox and delete the emails there.

See this article If a mailbox is offline in Mail on Mac - Apple Support

You are welcome. How did you lose emails? Have you checked at iCloud.com/Mail using a computer browser to see if they are there?