Preheat the 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 butter and caster sugar until pale and fluffy.
Beat in the eggs one at a time, mixing well after each addition.
Add the vanilla extract and mashed bananas to the mixture and stir until combined.
In a separate bowl, sift together the self-raising flour, baking powder, and salt.
Fold the dry ingredients gently into the wet ingredients until just combined, being careful not to overmix.
Pour the batter into the prepared cake tin and smooth the top with a spatula.
Bake in the preheated oven for 55 to 60 minutes, or until a skewer inserted into the center 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.