Preheat the oven to 350°F (175°C). Grease a 9×5 inch loaf pan or line it with parchment paper.
In a large mixing bowl, mash the ripe bananas with a fork until smooth.
Add the two large eggs to the mashed bananas and whisk until well combined.
Gradually add the self-rising flour to the banana and egg mixture. Stir gently until just combined; do not overmix.
Pour the batter into the prepared loaf pan and smooth the top with a spatula.
Bake in the preheated oven for 45 to 50 minutes, or until a toothpick inserted into the center comes out clean.
Remove the banana bread from the oven and allow it to cool in the pan for 10 minutes.
Transfer the bread to a wire rack to cool completely before slicing and serving.