Preheat the oven to 350°F. Grease and line two or three 8-inch round cake pans with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, beat the butter and sugar together until light and fluffy, about 2–3 minutes.
Add the eggs one at a time, mixing well after each addition.
Mix in the lemon zest, lemon juice, and vanilla extract.
Alternate adding the dry ingredients and buttermilk to the butter mixture, beginning and ending with the dry ingredients. Mix just until combined.
Toss the blackberries with 1 tablespoon of flour, then gently fold them into the batter.
Divide the batter evenly among the prepared pans and smooth the tops.
Bake for 30–35 minutes, or until a toothpick inserted in the center comes out clean.
Let the cakes cool in the pans for 10–15 minutes, then remove and cool completely on wire racks.