In a high-speed blender, add the peach, corn, cucumber, water, lime, oil, and salt and blend on high until smooth. Taste and add more salt or lime, if needed. Pour into a large bowl with a lid. Chill overnight or for at least 8 hours.
Right before serving, stir in most of the mix-ins (peaches, corn, and cucumber), saving some for topping (this just makes it look prettier, you can just add it all in if you want).
Scatter chopped avocado over top along with any peach, corn, and cucumber you saved for garnishing. Top with black pepper and drizzle of olive oil.