Yes, but technically you always could.

Because you fetch an invoice, that invoice contains the message. And now you can prove that you wrote the message, and that the recipient received the message: the invoice they sign has your message in it, and your key (often a made up one for that payment). 

 Interesting. Is there a limit to the message length? I'm thinking about using this for live streaming super chats. 

 The total length needs to fit in 64k, so yes. But the other fields tend to be no more than a few kB... 

 Thanks for the replies and all of the hard work!