My character can stop movement of the Death Star!

You should reply to Answers via comment, not by adding a new answer. Anyway, in order to help you we need a visualization of what's going on. Otherwise we're just throwing guesses out. Show us some screenshots, specifically your rigidbody/collider settings on the two objects.

As suggested below, if your Player is stopping the ship, it could be that the Player is set to being Kinematic. If that doesn't do it, try removing your Player script entirely and see if the ship can run it over. If that doesn't do it, it's almost definitely a setting in either your ship or the player Rigidbody or Collider.

Ok, I'll make a video. But I can't understand: You don't read what I wrote? How can I set player to kinematic if it has CharacterController (not rigidbody)? I repeat you: Ship - rigidbody, player - characterController. But get back to business; it would be ok if player could stop ship anyway, but there is one exception. If I'm standing on the floor, ship could fly up! Its the same thing as standing near wall and trying to fly forward, but if I'm standing near wall and trying to fly forward, then ship STOPS. Attention: don't look at the graphics while watching the video, I didnt work with it. ;)

Surely for the character to be effected by gravity you have to have both a rigidbody and the CharacterController attached to the player, so you should have a rigidbody on your character, so if you dont at the moment add one and change the masses of the rigidbodies etc

Rigidbody and character controller at the same object? o_O Sorry, its new for me. I used to think that there should be only one. Ok, I'll try to make hybrid, but what about issues with hybridisation? For example, I use function Move() of the CC and character also gets affected by forces...Anyway, thanks for note :)

http://www.youtube.com/watch?feature=player_detailpage&v=H8D3EB8eLDg

http://www.youtube.com/watch?v=NLanxu_8coQ&feature=mfu_in_order&list=UL

That's the video