Pre-heat the oven to 175℃ (350℉).
Line and/or grease an 18cm (7") springform (see note 3).
Add the butter to a pan and heat it until most of it is melted. Remove from the heat and set to one side for a moment.
In a bowl, whisk the eggs, sugar, vanilla extract, and lime (two tablespoons of juice and the zest) together.
Add in the melted butter and whisk just until everything is combined.
Sift in the flour and once more, just whisk until everything is combined.
Pour into the prepared cake pan and gently push the fresh blackberries into the batter (there's no need to completely submerge them).
Bake the cake in the lower part of the oven for 20 minutes. The top will still jiggle a little, but it will firm up.
Allow the cake to cool before adding the icing.