Don't know what county. I have seen one map that had the shore north of the city labeled Gotham County. Wayne Manor is outside the city on the mainland and is on the last few hundred acres of what used to be many square miles of Wayne land from when England captured the place from the Dutch.
Several maps are at
mapping Gotham . ON the second "No Man's Land" map, Wayne Manor is at 3 where the highway has a upside down T on the north side of the river. That article also links to
a tourist map of Gotham which explains where many of the names on the map come from.
