Does a winning strategy exist for Don't Break the Ice?
If I understand the game correctly, there should be a winning strategy for the player going last. If we first disregard the corners, then the bear will fall when the second last block is removed:

--X---
--X---
--BB--
--BB--
------
------


(or similar)
Since the number of blocks is even, this block will be removed by the player going first.
Now, there are two complication. First, the last block removed may include the corner:

-X----
-X----
-XBB--
--BB--
------
------


or

-X----
-XX---
--BB--
--BB--
------
------


Since the number of remaining blocks is now odd, the player going last would lose. 
Second, a block may fall due to being disconnected. As Steven notes, only four blocks can possibly fall this way. If an odd number of blocks fall, then the player going last will be the one to make the last move in the first setting. 

However, both of these can easily be avoided by attacking the corner blocks first. Since it requires two moves to remove a corner block, the second player can always remove the corner block before the first can make it fall, and of course before all side blocks are removed. This ensures that the final position will be of the first kind, and thus that the player going first loses.