Zest and juice the lemon. In a saucepan, stir together figs, lemon zest, lemon juice, sugar, and water. Bring to a boil. Turn heat down to simmer for 15 minutes, until most of the liquid cooks out.
Transfer to a food processor. Puree until smooth. Preheat oven to 170F. Line a half baking sheet with parchment. Using a spatula or offset spatula, spread out the fig puree into an even layer, about 1/8-inch thick.
Cook in the oven for 2 hours. (Top should be tacky—a little sticky and not completely dry). Cool completely.
Cut the parchment into long thin strips of fig fruit leather, then roll it up.