They can definitely fly.
At one time I had bantam Buttercups, AB's and Sumatras. I had to have a top on every pen. It was sheer chaos for the first few minutes whenever I would let them out to free-range. Thank goodness I had 5 acres for them to use.
Mine always got my blueberries and raspberries before I could.

Mine always got my blueberries and raspberries before I could.