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