To make simple syrup, in a small pot, add sugar and water. Cook over medium heat until the sugar has dissolved. Remove from heat and cool to room temperature. Transfer to an airtight container and store in the fridge for up to 1 week.
In a serving glass, crush the mint leaves with a muddler.
In a blender, blend the peaches until smooth. Pour the puree into the serving glass. Add simple syrup, lemon juice and white rum. Add ice and soda. Stir to mix. Garnish with extra mint leaves. Serve immediately.
Notes
You can use yellow peach, but you may want to adjust the sweetness, as yellow peach is slightly more tart.