Yes automatic transmission is not just possible in bikes, but there are actual bikes with automatic transmission.
For one there's the very famous MTT Turbine better known as Y2K, the Millennium bike. It has automatic transmission with two gears.
If you have heard of FJR1300, the 2006 model somes with automatic transmission. And there is the optional AE version, fitted with a clever automatic clutch and a remote shifter toggle on its left handlebar.
I'll post the pics of this bike.