Preheat oven to 350°F (175°C). Grease a loaf pan.
In a large bowl, mix together the mashed banana, grated zucchini, sugar, and eggs.
Add the vegetable oil and vanilla extract, and mix well.
In another bowl, combine the flour, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, stirring until just combined.
Pour the batter into the prepared loaf pan.
Bake for 60 minutes, or until a toothpick inserted into the center comes out clean.
Let the bread cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.