Preheat the oven to 350°F. Line a 9x9-inch baking pan with parchment, leaving an overhang for easy lifting and lightly grease the paper.
In a large bowl, mash the ripe bananas until mostly smooth. Add the peanut butter, maple syrup and vanilla, and whisk or beat by hand until the mixture is glossy and combined.
In a separate bowl stir together the rolled oats, ground flaxseed, cinnamon and salt. Toss the dry mix briefly so the spices are evenly distributed.
Fold the dry ingredients into the banana-peanut butter mixture until everything is well coated. Fold in about two-thirds of the chocolate chips, reserving the rest to sprinkle on top.
Transfer the batter to the prepared pan. Press the mixture firmly and evenly into the pan using a spatula or the back of a greased measuring cup; scatter the remaining chips on the surface and press lightly.
Bake for 20 to 22 minutes, until the edges are lightly golden and the center is set when gently pressed. Remove from the oven and cool on a wire rack for 15–20 minutes.
For neater slices and firmer bars, refrigerate the pan for 20–30 minutes before lifting the whole slab out on the parchment and cutting into 12 bars.