We present an algorithm for American option pricing based on stochastic approximation techniques. Besides working on a finite subset of the exercise dates (e.g. considering the associated Bermudean option), option pricing algorithms generally involve another step of discretization, either on the state space or on the underlying functional space. Our work, which is an application of a more general perturbed gradient algorithm introduced recently by the authors, consists in approximating the value functions of the classical dynamic programming equation at each time step by a linear combination of kernels. The so-called kernel-based stochastic gradient algorithm avoids any a priori discretization, besides the discretization of time. Thus, it converges toward the optimum of the non-discretized Bermudan option pricing problem. We present a comprehensive methodology to implement efficiently this algorithm, including discussions on the numerical tools used, like the Fast Gauss Transform, or Brownian bridge. We also compare our results to some existing methods, and provide empirical statistical results.