I want to install a dishwasher in an kitchen island. It is away from the sink, can I connect the dishwasher drain pipe to the existing drain pipe from sink after the "P" TRAP?

You connect DW discharge hose into sink drain BEFORE the trap. If you connect after the trap, you will get sewer get escaping into DW.

Depending on the plumbing Code in your area, there are 2 ways of installing DW: One is using Air Gap and the Other is to create high loop.

If you have Garbage Disposer, you connect DW into Disposer's side inlet. If you don't have Disposer, you connect into the drain via Branch "T".

First two drawings illustrate high loop installation, Second drawing illustrate installation using air gap

Hope that will help... Let us know how you did... Milo