The primary way to kill a Ninja Bear is with the Imperial Bot. Use the Bot on the Ninja Bear to turn it into a Tombstone.
However, Ninja Bears will also die if they have no possibility of moving (as do normal Bears), so a second way to kill them is by allowing the board to fill up--Ninja Bears move by jumping to empty squares, so if there are no empty squares then they can't move and will die. Ordinarily a full board means Game Over; however, if the Tombstones of the dead bears make a Church then some space will be freed up and you can continue playing.
In the picture below, the screenshot on the left shows that I'm about to place a grass in the bottom right, just above the Mansion. This fills the board. Since none of the normal Bears or Ninja Bears can move, they all die and turn into Tombstones. Since there are more than three Tombstones, the Tombstones combine to form a Church and I'm able to continue playing rather than getting a Game Over message.