Line the bottom of an 8½-by-4½-inch loaf pan with parchment or wax paper.
Stir together the cookies and butter in a small bowl until moistened.
Remove the chocolate ice cream from the freezer and let sit until softened slightly, about 5 minutes. Scoop the chocolate ice cream into a medium bowl and stir until softened and spreadable. Spread the ice cream evenly in the bottom of the prepared pan with an offset spatula or spoon. Sprinkle with a third of the cookie mixture and freeze until firm, 25 to 30 minutes. Repeat this process with the vanilla and strawberry ice creams and the remaining cookie mixture. Freeze, covered, until firm, at least 2 hours and up to 2 days.
When the cake is ready to serve, dip an offset spatula or thin knife into hot water and run it around the edge of the loaf pan to loosen the cake. Invert it onto a serving plate and remove the parchment.