Memristive materials play a key role in the development of neuromorphic technology given that they can combine information processing with volatile or nonvolatile memory storage in a single computational component. Both functionalities are strictly required for the design and implementation of neuromorphic circuits. Many of these bioinspired materials emulate the characteristics of memory and learning processes that happen in the brain. The memristive properties of a two-terminal (2-T) organic device based on ionic migration mediated by an ion-transport polymer are reported here. The material possesses unique memristive properties: it is reversibly switchable, shows tens of conductive states, presents Hebbian learning demonstrated by spiking time dependent plasticity, and behaves with both short- and long-term memory in a single device. The origin and synergy of both learning phenomena are theoretically explained by means of the chemical interaction between ionic electrolytes and the ion-conductive mediator. Further discussion on the transport mechanism is included to explain the dynamic behavior of these ionic devices under a variable electric field. This polymer-based composite as an outstanding neuromorphic material is proposed for being tunable, cheap, flexible, easy to process, reproducible, and more biocompatible than their inorganic analogs.