We study the baby Skyrme model in (2+1)-dimensions built on a finite cylinder. To this end, we introduce a consistent ansatz which is able to reduce the complete set of field equations to just one equation for the profile function for arbitrary baryon charge. Many analytic solutions both with and without the inclusion of the effects of the minimal coupling with the Maxwell field are constructed. The baby skyrmions appear as a sequence of rings along the cylinder, leading to a periodic shape in the baryon density. Linear stability and other physical properties are discussed. These analytic gauged baby Skyrmions generate a persistent U(1) current which cannot be turned off continuously as it is tied to the topological charge of the baby Skyrmions themselves. In the simplest non-trivial case of a gauged baby Skyrmion, a very important role is played by the Mathieu equation with an effective coupling constant which can be computed explicitly. These configurations are a very suitable arena to test resurgence in a non-integrable context.