Through imaging the same spatial area by hyperspectral sensors at different spectral wavelengths simultaneously, the acquired hyperspectral imagery often contains hundreds of band images, which provide the possibility to accurately analyze and identify a ground object. However, due to the difficulty of obtaining sufficient labeled training samples in practice, the high number of spectral bands unavoidably leads to the problem of a "dimensionality disaster" (also called the Hughes phenomenon), and dimensionality reduction should be applied. Concerning band (or feature) selection, conventional methods choose the representative bands by ranking the bands with defined metrics (such as non-Gaussianity) or by formulating the band selection problem as a clustering procedure. Because of the different but complementary advantages of the two kinds of methods, it can be beneficial to use both methods together to accomplish the band selection task. Recently, a fast density-peak-based clustering (FDPC) algorithm has been proposed. Based on the computation of the local density and the intracluster distance of each point, the product of the two factors is sorted in decreasing order, and cluster centers are recognized as points with anomalously large values; hence, the FDPC algorithm can be considered a ranking-based clustering method. In this paper, the FDPC algorithm has been enhanced to make it suitable for hyperspectral band selection. First, the ranking score of each band is computed by weighting the normalized local density and the intracluster distance rather than equally taking them into account. Second, an exponential-based learning rule is employed to adjust the cutoff threshold for a different number of selected bands, where it is fixed in the FDPC. The proposed approach is thus named the enhanced FDPC (E-FDPC). Furthermore, an effective strategy, which is called the isolated-point-stopping criterion, is developed to automatically determine the appropriate number of bands to be selected. That is, the clustering process will be stopped by the emergence of an isolated point (the only point in one cluster). Experimental results on three real hyperspectral data demonstrate that the bands selected by our E-FDPC approach could achieve higher classification accuracy than the FDPC and other state-of-the-art band selection techniques, whereas the isolated-point-stopping criterion is a reasonable way to determine the preferable number of bands to be selected.