If the mail app you are using allows you to print, then you can save emails as PDFs.
1) Select the email
2) Choose Print from the menu.
3) Pinch out on the preview portion of the print dialog. This will expand the email to a full screen preview.
4) Tap the share icon (box with up arrow)
5) Select the Save To Files option
6) Choose Dropbox as the destination (Must have DropBox set up in Files)
This works with both the Mail app and the Gmail app. I haven’t tried it with other apps or web email clients.
If all you want is the body of the email, just select it and use the same procedure for PDF, or use the Share option in the pop-up menu to save it as text.
You can get fancier with the text selection options and the Shortcuts app. However, the only way to save the entire email, including the header, is saving it as a PDF as described above.
If all you want to do is save the attachment, then select it (long press), use the share option in the pop-up menu, and Save To Files. That’s the Mail app. Not sure how to do in in the Gmail app, since I don’t have any attachments there at the moment, but it should be similar.