Here is the full list of all 14 Battle Islands trophies - there are 11 bronze, 2 silver and 1 gold.
Upgraded the supply depot to rank 10
Attained 45 stars in the campaign
Upgrade the command bunker to rank 8
Loot 2500000 supplies from the enemy
Promote the bomber in the command bunker
Attain 100 offensive victories against the enemy
Garrison 500 army units
Expanded your base to 4 islands
Promote the warship in the command bunker
Remove 8 obstacles from your islands
Attained 10 consecutive victories against the enemy
Lost 500 rifleman in battle
Complete the tutorial
Acquired 2500 dog tags from the enemy