Preheat oven to 180C/160C fan/gas 4. Butter and line the base and 2 long sides of a 10cm x 22cm (base measurement) loaf pan with baking parchment.
Combine flour, coconut, lime rind and sugar in a large bowl. Whisk the melted butter, eggs and coconut milk in a small bowl. Add to the flour mixture. Stir to combine and whisk until lump free. Pour into the prepared tin. Smooth the surface.
Bake for 45 mins or until a skewer inserted in the centre comes out clean. Set aside in the pan for 5 mins. Turn onto a wire rack to cool completely.
To make the lime drizzle icing, place the icing sugar in a bowl. Add enough lime juice to form a runny paste. Pour over the top of the cooled cake.