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