Unfortunately, the options for forwarding and backing up emails are severely limited in Apple’s Mail app on iOS. The e-mails can certainly neither be shared via a messenger, nor can they be saved on the device. At least a cursory glance at the app makes this look. Apple has integrated both functions. However, they are not quite as intuitive to find and use as you are used to from Apple.

How to save emails as PDF – with 3D Touch
Pick up your iPhone (iPhone 6s to iPhone X only) and open the Mail app. Select the email you’d like to save as a PDF. Now tap on the action button at the bottom with which you can forward, reply to or print emails. Select the “Print” item here.

You will now see a small preview. First press it lightly to see the familiar “3D Touch” preview. Keep your finger on it. Now press a little harder to open the view. With the new view, you will also see the share button. Tap on it . Select files, cloud storage or iBooks to save the email as a PDF. Alternatively, you can add the email to your notes or send it via one of your installed messaging services – the choice is yours!
How to save e-mails as PDF on the iPhone and iPad – without 3D Touch
If you don’t have an iPhone with 3D Touch or an iPad, you can still save emails as PDF. Follow the instructions above to preview. Instead of pressing firmly on it, pull the preview large with two fingers as you would with zoom. You can then save the document as a PDF on your device or cloud storage using the share button.