Combine coconut milk, sugar and ginger in a small saucepan. Bring to a simmer over medium heat. Simmer for 1 1/2 minutes. Remove from heat and let stand for 20 minutes. Strain into a large bowl.
Place pineapple in a food processor and process until smooth. Add the pineapple puree to the coconut milk mixture and whisk until blended. Whisk in lime juice. Cover and refrigerate until chilled, about 1 hour.
Pour the pineapple mixture into an ice cream maker and freeze according to manufacturer's directions. (Alternatively, freeze mixture in a shallow metal pan until solid, about 6 hours. Break into chunks and process in a food processor until smooth.)
Serve immediately or transfer to a storage container and let harden in the freezer for 1 to 1 1/2 hours. Serve in chilled dishes.