Preheat oven to 350 degrees F (175 degrees C). Line an 8-inch square pan with parchment paper.
Beat butter and sugar in a bowl with an electric mixer until pale and fluffy, about 3 minutes. Add whole eggs and egg yolk one at a time, beating well after each addition. Mix in lychee syrup and rose water until thoroughly combined, 2 to 3 minutes.
Whisk flour, baking powder, and salt together in a bowl. Add half of the flour mixture to the butter mixture. Beat until flour is thoroughly mixed in before adding second half. Beat until batter is smooth, about 30 seconds. Gently fold in half of the lychees.
Pour batter into the prepared pan. Spread remaining lychees evenly on top of the batter.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 40 to 45 minutes. If edges brown too quickly, cover pan with aluminum foil. Place pan on a cooling rack to cool cake.