The ability to learn a policy for a sequential decision problem with continuous state space using on-line data is a long-standing challenge. This paper presents a new reinforcement-learning algorithm, called iKBSF, which extends the benefits of kernel-based learning to the on-line scenario. As a kernel-based method, the proposed algorithm is stable and has good convergence properties. However, unlike other similar algorithms,iKBSF's space complexity is independent of the number of sample transitions, and as a result it can process an arbitrary amount of data. We present theoretical results showing that iKBSF can approximate (to any level of accuracy) the value function that would be learned by an equivalent batch non-parametric kernel-based reinforcement learning approximator. In order to show the effectiveness of the proposed algorithm in practice, we apply iKBSF to the challenging three-pole balancing task, where the ability to process a large number of transitions is crucial for achieving a high success rate.