How do I stop my island from sliding across the floor?
There are a few options...  You can remove the non-marring plastic feet provided by Ikea, and replace them with some rubber nail or screw on feet, or you can get rubber "caster cups" that the existing feet can set on.  Rubber will provide protection between the floor and the table, but will stop if from sliding.

You should be able to find both and any home improvement store.

Screw on rubber feet (found on Amazon):



Rubber caster cups (image from lowes.com)