Preheat your oven to 180ºC (356ºF) and line an 8-inch round baking pan with parchment paper. If using a non-stick pan, grease it with melted butter and a light dusting of flour for a smooth cake.
In a medium mixing bowl, combine flour, baking powder, and salt. Set aside.
In a large mixing bowl, use electric hand beaters or a stand mixer with the paddle attachment. Start with softened butter, beating until it turns creamy. This eases the addition of sugar.
Add sugar and lemon zest, beat for 2 minutes until pale and creamy, scraping the sides as needed.
Add eggs one at a time, beating for 10 seconds after each addition. Remember to scrape the sides.
Pour in milk, oil, lemon juice, and vanilla. Beat for 15 seconds until well combined, scraping the sides.
Add half of the dry flour mixture and beat until just combined. Then, add the remaining flour and beat until incorporated, being careful not to over-beat. Use a spatula to gently fold in any excess flour on the sides.
Gently fold in blackberries, making sure not to over-mix. Pour the batter into the lined pan and smooth the top with a spatula.
Press a few extra blackberries on top of the cake batter.
Bake for 45-50 minutes, until a skewer inserted into the middle comes out clean and it's lightly golden all over.
Remove from the oven and let it cool in the pan for 15 minutes.
Carefully transfer the cake to a wire rack to cool completely.
If desired, dust with a light layer of icing sugar before serving.
Serve and savor the delight!