The local governor said the military was repelling an attack. He uploaded video showing a large fire while an explosion could also be heard.
An unconfirmed Russian media report says four Ilyushin 76 transport planes were damaged at the airport.
Pskov is more than 600km (372 miles) away from Ukraine, close to the border with Estonia.
In recent weeks Ukraine has increased its use of explosive drones to attack targets inside Russia.