Well, inline chat, like IRC is generally a single-line input... though I suppose you could use a textarea rows="1" and strip \n on paste, etc, that wouldn't be too bad as you're likely listening for \n to post the message anyway.
Either way, not using a typical name/id for the text input would probably work better in terms of not getting suggestions...
Chat input is typically single-line, submit-on-enter. Messaging apps seem to be transitioning to optional multiline but the default action for enter still seems to be submit so it's perfectly reasonable to use <input> for chat input.
That would indeed be the norm for forums. But not for chat. Real time chat is typically more usually a single line entry. Having auto-complete on that is irritating and utterly spoils the UI.
And <textarea> doesn't auto-complete regardless of explicit options.