No, the 1/1's live.
The 2 1/1's assign first strike damage before the 2/2. The 2 damage from the 1/1's kills the 2/2 before it has a chance to deal any damage back.
the 1/1 stay alive
502.2b At the start of the combat damage step, if at least one attacking or blocking creature has first strike or double strike, creatures without first strike or double strike dont assign combat damage. Instead of proceeding to end of combat, the phase gets a second combat damage step to handle the remaining creatures. In the second combat damage step, surviving attackers and blockers that didnt assign combat damage in the first step, plus any creatures with double strike, assign their combat damage.
Combat damage is assigned like this
but we all know bears are AMAZING so your arguement is invalid grin and BEAR it