Here is the full list of all 28 The Islander achievements.
Cut down a total of 250 Trees
Break a total of 250 Rocks
Reach Level 20 in a Single Run
Buy a total of 200 Cocobots
Catch a total of 100 Parrots
Catch a total of 100 Starfish
Catch a total of 100 Fish
Have at least 1M Gold
Buy a total of 250 Animals
Buy a total of 100 Animals
Plant a total of 250 Trees
Plant a total of 250 Crops
Plant a total of 250 Bushes
Plant a total of 100 Bushes
Build a total of 750 Decorations
Plant a total of 100 Trees
Plant a total of 100 Crops
Buy a total of 25 Animals
Plant a total of 25 Trees
Have at least 100K Gold
Reach Level 10 in a Single Run
Buy a total of 100 Land Expansions
Build a total of 250 Decorations
Plant a total of 25 Bushes
Reach Level 5 in a Single Run
Plant a total of 25 Crops
Have at least 1000 Gold
Build a total of 100 Decorations