A bandwidth selector for local polynomial fitting is proposed following the bootstrap idea, which is just a double smoothing bandwidth selector with a bootstrap variance estimator, defined as the mean squared residuals of a pilot estimate. No simulated resampling is required in this context, since the needed expressions can be calculated explicitly. A simple, iterative data-driven procedure is proposed to estimate the variance and the bandwidth. A simulation study shows that this bandwidth selector performs very well, and it performs uniformly better than a double smoothing bandwidth selector using a difference-based variance estimator. The above mentioned bootstrap variance estimator is also a side result of this paper. It performs clearly better than the difference-based one. In a test example, the averaged squared error of this estimator in 500 replications achieved the theoretical lower bound already with a sample size of only n = 200.