1 1/2 pounds fresh or frozen blackberries (thaw if frozen)
1/3 cup granulated sugar
2 serrano peppers, stemmed and coarsely chopped
Pinch fine salt
2 teaspoons freshly squeezed lime juice
Arrange a rack in the middle of the oven and heat to 170°F. Line a rimmed baking sheet with a silicone baking mat and place it on a flat work surface; set aside.
Set a fine-mesh strainer over a medium saucepan; set aside. Place the blackberries, sugar, serranos, and salt in a blender and blend on high speed until very smooth, about 1 minute. Pour the mixture through the strainer into the saucepan and, using a rubber spatula, scrape against the inside surface of the strainer to push the purée through until only mostly seeds remain. Remove the strainer (make sure to scrape any purée stuck to the underside of the strainer into the pan) and discard the seeds.
Heat the purée over medium heat and cook, stirring occasionally with a rubber spatula and scraping the sides and bottom of the pan, until the mixture starts to simmer around the edges. Reduce the heat to medium-low and simmer, stirring occasionally, until the mixture thickens slightly and reduces to about 1 3/4 to 2 cups, about 20 minutes. Remove the pan from the heat and stir in the lime juice. Transfer the mixture to a heatproof container with a spout (like a liquid measuring cup or cleaned blender pitcher).
Slowly pour the mixture onto the baking mat, tracing the inside of the colored border to create a rectangle. (If your baking mat has no border, leave a 1-inch border from the edge.) Pour the remaining mixture within the borders of the rectangle in a zigzag pattern (do not pour it all into the middle of the baking mat). Using the rubber or an offset spatula, spread the mixture to cover any empty parts within the rectangle (the surface will not be even).
Keeping the baking sheet on a work surface, grasp the edges of it and press against the uncovered border of the baking mat with your thumbs. Gently shake the sheet back and forth to even out the surface of the mixture, rotating the baking sheet and shaking as needed.
Bake until the surface is slightly sticky to the touch but, when pressed in several different places, a finger does not leave an indentation, about 4 to 7 hours. Remove the baking sheet from the oven, place on a wire rack, and let cool completely.
Cut a 16-inch-long piece of wax or parchment paper. Starting at one short edge of the cooled fruit leather, pull it up from the baking mat and transfer it to the paper. Using kitchen scissors, cut through both the fruit leather and paper to form the desired-size strips or pieces. Roll them up, paper and all (to prevent sticking). Store in an airtight container at room temperature for up to 2 weeks.