Here is the full list of all 96 Battle Islands achievements.
Defend against 10 attacks
Garrison 25 naval units
Promote the landing craft to rank 4
Garrison 25 army units
Acquire 1100 dog tags from the enemy
Attain 5 defensive victories against the enemy
Lose 500 riflemen in battle
Expand your base to 4 islands
Promote the bomber in the command bunker
Attain 30 stars in the campaign
Promote the mortarman to rank 3
Concede 5 defensive defeats against the enemy
Upgrade command bunker to rank 8
Promote the submarine in the command bunker
Attain 3 consecutive victories against the enemy
Promote the rifleman to rank 5
Promote the warship to rank 2
Promote the fighter to rank 2
Promote the tank to rank 2
Promote the gunboat to rank 2
Promote the jeep to rank 2
Promote the tank in the command bunker
Promote the fighter in the command bunker
Promote the warship in the command bunker
Upgrade a supply depot to rank 5
Upgrade command bunker to rank 5
Loot 50000 supplies from the enemy
Promote the rifleman to rank 3
Promote the landing craft to rank 2
Attain 15 stars in the campaign
Promote the jeep in the command bunker
Remove 8 obstacles from your islands
Remove 4 obstacles from your islands
Promote the mortarman in the command bunker
Promote the gunboat in the command bunker
Upgrade command bunker to rank 3
Upgrade a supply depot to rank 2
Lose 50 riflemen in battle
Capture your first island
Conquer the first campaign map
Complete the tutorial
Expand your base to 2 islands
Promote the torpedo bomber in the command bunker
Defend against 250 attacks
Defend against 5000 attacks
Lose 10000 riflemen in battle
Garrison 500 army units
Garrison 5000 army units
Garrison 500 naval units
Garrison 5000 naval units
Garrison 25 air force units
Garrison 500 air force units
Garrison 5000 air force units
Upgrade a supply depot to rank 10
Attain 45 stars in the campaign
Remove 16 obstacles from your islands
Loot 2500000 supplies from the enemy
Loot 100000000 supplies from the enemy
Acquire 1500 dog tags from the enemy
Acquire 2500 dog tags from the enemy
Expand your base to 6 islands
Attain 5 offensive victories against the enemy
Attain 100 offensive victories against the enemy
Attain 1000 offensive victories against the enemy
Concede 5 offensive defeats against the enemy
Concede 100 offensive defeats against the enemy
Concede 1000 offensive defeats against the enemy
Concede 100 defensive defeats against the enemy
Concede 1000 defensive defeats against the enemy
Attain 10 consecutive victories against the enemy
Attain 25 consecutive victories against the enemy
Promote the rifleman to rank 8
Promote the mortarman to rank 5
Promote the mortarman to rank 8
Promote the jeep to rank 4
Promote the jeep to rank 6
Promote the tank to rank 4
Promote the tank to rank 6
Promote the gunboat to rank 4
Promote the gunboat to rank 6
Promote the warship to rank 4
Promote the warship to rank 6
Promote the submarine to rank 2
Promote the submarine to rank 4
Promote the submarine to rank 6
Promote the fighter to rank 4
Promote the fighter to rank 6
Promote the bomber to rank 2
Promote the bomber to rank 4
Promote the bomber to rank 6
Promote the torpedo bomber to rank 2
Promote the torpedo bomber to rank 3
Promote the torpedo bomber to rank 4
Promote the landing craft to rank 6
Attain 100 defensive victories against the enemy
Attain 1000 defensive victories against the enemy