Tilt the serving glass to one side and drizzle about 1 teaspoon of coconut milk horizontally along the inside of the glass. Tilt the glass in the other direction and drizzle another teaspoon of coconut milk horizontally along the inside of the glass. Set aside.
In a blender, add the strawberries, lime juice, lemon-lime soda and ice cubes. Blend on the smoothie setting.
Add the rum and pulse several times.
Pour the strawberry mixture into the serving glass.
Dip the pointed end of a wooden skewer into the remaining coconut milk and draw a swirl on top of the strawberry daiquiri with it. Continue to dip the skewer in the coconut milk to draw the desired swirls on top of the daiquiri.