Preheat your oven to 340°F (170°C).
In a small bowl, mix together the dairy-free milk and apple cider vinegar. Set it aside for a few minutes to thicken and curdle.
In a large bowl, combine the thickened milk mixture with the peach juice, yogurt, and olive oil.
Sift in the self-rising flour, baking powder, and baking soda. Gently mix everything until just combined.
Pour the batter into three 6-inch round baking pans. Bake for 20-30 minutes, or until the cakes are well-risen and a toothpick inserted into the center comes out clean.
While the cakes are cooling, whip the dairy-free whipping cream with the powdered sugar and vanilla extract using a stand mixer or an electric whisk until soft peaks form.
Once the cakes are completely cooled, layer them with the whipped cream and top with fresh peach slices.