Is there any bicycle that can fly?
The first human-powered, fixed-wing aircraft flew in 1961, the SUMPAC. Several other such craft were flown in the 60's and 70's, including the Gossamer Condor, which won the prestigious Kremer Prize for human-powered flight. All of these were powered by a human pedaling, like on a bicycle. 

An aircraft with wings that flap like a bird's is called an Ornithopter. None that I can find were capable of getting into the air on their own. Early, unmanned ornithopters were powered by large rubber bands or gunpowder charges. The first verified human-powered ornithopter flew in 1929, but it too was towed into the air.