Preheat your oven to 180°C (350°F) and grease a 20cm (8 inch) round cake tin, then line the base with baking parchment.
In a large mixing bowl, cream together the softened unsalted butter and caster sugar until pale and fluffy using an electric mixer or wooden spoon.
Add the eggs one at a time, beating well after each addition to fully incorporate them into the mixture.
Stir in the vanilla extract.
Sift the self-raising flour, baking powder, and a pinch of salt into the bowl and gently fold into the wet ingredients.
Fold in the mashed ripe bananas until the batter is smooth and evenly combined.
Pour the batter into the prepared cake tin and smooth the top with a spatula.
Bake in the preheated oven for 40 to 45 minutes, or until a skewer inserted into the center of the cake comes out clean.
Remove the cake from the oven and allow it to cool in the tin for 10 minutes.
Turn the cake out onto a wire rack to cool completely before serving.