Arrange a rack in the middle of the oven and heat to 350°F. Line a baking sheet with parchment paper.
Mash the banana with a fork in a medium bowl until smooth, then stir in the coconut. Scoop 1 tablespoon-portions onto the baking sheet, spacing them evenly apart. Bake until the edges of the cookies are golden-brown, 15 to 17 minutes. Place the pan on a wire rack and cool completely.
Place the chocolate chips and coconut oil in a small, microwave-safe bowl. Microwave on high until the chocolate is melted, 45 seconds to 1 minute.
Dip the bottom of each cookie into the chocolate and place back on the parchment-lined baking sheet. Transfer any remaining chocolate into a small zip-top bag. Snip the corner of the bag off and drizzle the chocolate over the tops of the cookies. Refrigerate until the chocolate is set, about 15 minutes.
