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In this paper we consider the one-dimensional, biased, randomly trappedrandom walk when the trapping times have infinite variance. We prove sufficientconditions for the suitably scaled walk to converge to a transformation of astable L\'{e}vy process. As our main motivation, we apply subsequentialversions of our results to biased walks on subcritical Galton-Watson treesconditioned to survive. This confirms the correct order of the fluctuations ofthe walk around its speed for values of the bias that yield a non-Gaussianregime.

Non-Gaussian fluctuations of randomly trapped random walks