No, as you suspect this is not physically allowed, since this would constitute a violation of the principle of [special relativity](). There is a somewhat simpler similar situation that helps to explain why this is. Imagine you have a pair of scissors with very long blades, which you closed really fast. When the blades are short, the speed with which the ends of the scissors will move will be equal to wR, where w is the angular velocity (the speed with which you are reducing the angle the scissors make), and R is the length of the blades as measured from the pivot. You may then think that you could just make the length enough and at a certain speed the end of the blade would move at the speed of light, but this is not so.

To see why this is impossible, consider what is going on at a microscopic scale. When you press on the scissors, your hand only comes into contact with the atoms making up its surface of the handle. Those atoms in the handle will turn will push on their neighbors, and so on. Essentially what you are doing is you are creating a mechanical wave. This wave will then propagate through the blade at a speed determined by the density and compressibility of the material it is made of and will be similar to the speed of sound in that medium, which for realistic materials is far smaller than the speed of light.

Exactly the same thing will happen to the helicopter blades. Just as with the scissors, if you make the blade long enough, then "kick" caused by the movement of the rotor will not propagate instantaneously but will only propagate at a finite speed towards the end of the blades. This will cause the blades to bend, such that the tip will always move (much) slower than the speed of light.