As an alternative approach, you could convert the raster to a polygon, then edit the polygon to remove the un-wanted regions/islands. Then, you could use this new polygon as a mask.
Raster to polygon:
Convert the raster to polygon with the raster to polygon tool.
Uncheck the 'simplify polygons' option so that the polygon edges in the output conform to exact raster cell edges.
Then open the editor tool and the attribute table of the new polygon. Use the select tool to identify the islands/regions that you don't want to keep, and delete them from the attribute table. You can quickly identify the areas you don't want in the attribute table by first identifying the areas that you do want.
Extract by mask:
Now that you have a masking layer, use the extract by ask to remove that raster cells from your analysis.