The first thing you will have to do is to get a chair.
You can find it at the blueprints-furniture menu and you need a car spot.
You can buy it for $400. Now, we have to move the car next to the chair. Must be aligned on the same line.
After that, you will need to get an axe in order to cut the palm.
Now, sit on a chair and respawn the car while sitting.
Now, you and the car are one body.
You can use this for jumping in the water.
This island is closer, choose this one to go.
You can breathe when you keep on jumping. There will be no damage.
Be patient and careful when you are cutting the palm tree.
Now this is the longest part, moving the wood to the base.
You have to keep jumping until you are in full health after every morning.
Reset the chair and finally you will get a big palm tree.