Do you have the space to plan a peninsula rather than an island? These are often more workable when the space is tight. It would not be very safe, and in some areas not to code to have an island with a cooktop and no surrounding countertop.

You absolutely cannot have a stove with no counter that is just looking for a disaster to happen. Could you please send a floor plan with measurements and some pictures of space too would be very helpful. In a 12 x10 kitchen you do not have room for an island. This picture shows what you might be able to do but as you see the stove can not go in the peninsula either.