Cut each banana in half lengthwise and halve each piece crosswise. Work with 1 wrapper at a time; cover the rest with a damp paper towel. Lay 1 wrapper on a work surface and lightly brush the edge with beaten egg white. Place a banana quarter 2 inches from the bottom of the wrapper and top it with 4 chocolate chips and 1 teaspoon of coconut. Fold the bottom edge over the filling and roll up 1 complete turn. Fold in the sides and continue rolling to form a tight cylinder. Repeat with the remaining ingredients.
In a medium saucepan, heat 2 inches of vegetable oil to 350°. Add 2 spring rolls at a time and fry until golden brown, about 1 minute per side. Drain on paper towels. Dust the spring rolls with the sugar and cocoa and serve.