Add the peaches, lime, carrots, honey, coconut water and ginger to a blender. Blend for a couple of minutes, until completely smooth.
Pour the mixture through a fine-mesh strainer into a large bowl or pitcher. Use the back of a spoon or your hands to really press and squeeze the mixture, getting out as much of the juice as possible.
For each drink, add 1/2 peach and a few mint leaves to the bottom of a glass. Muddle them together.
Fill the glass with ice, then pour in about 3/4 of the peach-carrot juice. Top with sparkling water for some fizz, or coconut water for a softer, naturally sweet finish.
Give it a stir, add a little fresh mint if you like.