In a medium pot over medium heat, bring the peaches, chili, sugar and water to a simmer for 5 minutes until the sugar has dissolved and the peaches have softened.
Transfer the contents to the jar of a blender and puree on high until you have a smooth peach puree. Set aside to cool.
When the peach puree has cooled, pour it into a pitcher filled with ice.
Add the lemon juice and the water and stir well.
Keep chilled and pour over ice. Garnish with the peach and lemon slices to serve.