Narrowband spikes have been observed in solar flares for several decades. However, their exact origin is still discussed. To contribute to understanding of these spikes, we analyze the narrowband spikes observed in the 800-2000 MHz range during the impulsive phase of the 2013 November 7 flare. In the radio spectrum, the spikes started with typical broadband clouds of spikes, and then their distribution in frequencies changed into unique, very narrow bands having noninteger frequency ratios. We successfully fitted frequencies of these narrow spike bands by those, calculating dispersion branches and growth rates of the Bernstein modes. For comparison, we also analyzed the model where the narrow bands of spikes are generated at the upper-hybrid frequencies. Using both models, we estimated the plasma density and magnetic field in spike sources. Then, the models are discussed, and arguments in favor of the model with the Bernstein modes are presented. Analyzing frequency profiles of this spike event by the Fourier method, we found the power-law spectra with the power-law indices varying in the -0.8 to -2.75 interval. Because at some times this power-law index was close to the Kolmogorov spectral index (-5/3), we propose that the spikes are generated through the Bernstein modes in turbulent plasma reconnection outflows or directly in the turbulent magnetic reconnection of solar flares.