Preheat the oven to 350℉. In a small bowl, whisk together the flour, baking soda, and salt. In a separate bowl, stir together the bananas and Greek yogurt.
In a large bowl, cream the butter and sugars on medium speed with an electric hand mixer or stand mixer. Beat until fluffy, about 2 minutes. Scrape down the bowl and add the eggs; mix until combined.
Add the flour mixture and mix on low until incorporated, then mix in the banana and yogurt mixture and mix on low until well combined, scraping down the bowl if needed. Gently fold in the blackberries.
Grease a standard-size loaf pan and line it with a parchment sling, then pour in the batter. Let it rest for up to ½ hour if using gluten-free flour.
Bake the loaf for 60-75 minutes, until set and a tester inserted in the middle comes out clean. Cool the loaf in the pan for 30 minutes, then gently place on a wire rack to cool completely. If you're using gf flour, the loaf will be soft but will set as it cools.
