For the cake: Place an oven rack in the center of the oven. Preheat the oven to 350 degrees F. Butter and flour a 8-by-8-inch glass baking dish.
Whisk the flour, baking soda, salt, baking powder and cinnamon in a medium bowl until blended. Beat the sugar, oil, eggs and vanilla in a separate large bowl to blend. Stir in the chopped peaches. Gradually add the dry ingredients and stir until blended. Pour the batter into the prepared baking dish. Bake until a tester inserted into the center of the loaf comes out with no crumbs attached, about 1 hour 10 minutes. Transfer the pan to a wire rack and cool for 15 minutes. Remove the loaf from the pan and cool completely, about 1 hour.
For the frosting: Beat the butter, cream, 2 tablespoons water and vanilla in a large bowl, using an electric hand mixer on medium speed until light and fluffy. Gradually beat in the powdered sugar until light and creamy, adding extra water, 1 teaspoon at a time, if needed. Spread the frosting over the top of the cooled loaf. Allow the frosting to set for 25 minutes. Cut the loaf into slices and serve.