Yes, but that self-reference happens outside the formal language, not within it. Gödel encoded formal statements in the language as integers, and thus turned the question of provability into statements of arithmetic. If the formal system is powerful enough to let you reason about arithmetic, you can thus trick the system to be self-referential, but only via this mechanism external to the system itself.
The thing is, as far as we have to deal with in the real world, there is nothing such as an absolutely isolated system. To my mind, self-reference only occurs in an interpretation, thus in the frame of some actions, which are only possible in a world where changes happen, most likely changing the frame of interpretation itself.
