Pour lychees into a colander over a bowl. Measure syrup and reserve 1/4 cup. Slice about 1/3 of lychees in half.
Put 1 cup raspberries in a pitcher and muddle them to release juices. Add all lychees and reserved syrup. Slowly pour in Prosecco. Stir with a long spoon to mix.
To serve each cocktail, add 1 or 2 of remaining raspberries, plus a lychee from pitcher, into a sparkling-wine flute, then pour in cocktail.