If you don't want a submit button to get in the way, then I'd recommend making it just an icon (like most text messaging apps). You can also only show it when the user focuses on the input box to simplify the initial UI even more.

However, we don't really know the context for why a submit button would get in the way of your UI. If you can fit it in, then do it because of all the UX and accessibility goodness you get with it. If not, it's not the end of the world and I'm sure any user would be able to figure out to press enter. If you're able to show a small hint somewhere to let them know, that'd be best.

A big reason to consider a submit button would also be the fact that some data being passed by a form field may need to be submitted after adding multiple lines to the textarea or input field.

Enter is used to create a new line to type on. If "Enter" is used exclusively to submit each line of text that has been typed - think of how many transactions this will multiply into, simply because you haven't allowed the user to do so without submitting each line individually.

People are used to seeing some sort of submit, even in chats apps. Removing it might be confusing. Not overwhelmingly so, but still...

NOTHING good for UX is EVER bad for your UI. If you cannot fit a button that says "send" there must be something really jacked up with your design.

It depends on the setting. If it's a formal one, where content is longer and needs more review before send, I'd have a button. If it's an actual chat application, enter should be fine. LinkedIn has both options, and features a toggle function for preference, and it doesn't take up very much space.

Enter + Submit button, my opinion. Its easier for all people, especially for those who don't use PC daily.