Preheat your oven to 350°F (175°C). Grease and flour a 9-inch cake pan.
In a mixing bowl, whisk together the flour, sugar, baking powder, baking soda, and salt.
Add the softened butter and mix until the mixture resembles coarse crumbs.
In another bowl, whisk together the eggs, vanilla extract, buttermilk, and lemon zest.
Gradually add the wet ingredients to the dry ingredients, mixing until just combined.
Gently fold in the blackberries.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 35-40 minutes, or until a toothpick inserted into the center comes out clean.
Let the cake cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.