Preheat oven to 350 degrees F (175 degrees C).

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Grease and flour a 2-quart baking dish.

Whisk sweetened condensed milk, sugar, eggs, and vanilla extract in a large bowl. Stir in whole kernel corn, cream-style corn, and butter.

Whisk cornmeal, flour, baking soda, and salt in a separate bowl; stir cornmeal mixture into the corn mixture.

Pour batter into the prepared baking dish.

Bake until a toothpick inserted into the center of the cornbread comes out clean, about 45 minutes. Serve warm.