The decrease in the widths of spectral bands in hyperspectral imaging leads to a decrease in signal-to-noise ratio (SNR) of measurements. The decreased SNR reduces the reliability of measured features or information extracted from hyperspectral images (HSIs). Furthermore, the image degradations linked with various mechanisms also result in different types of noise, such as Gaussian noise, impulse noise, deadlines, and stripes. This article introduces a fast and parameter-free hyperspectral image mixed noise removal method (termed FastHyMix), which characterizes the complex distribution of mixed noise by using a Gaussian mixture model and exploits two main characteristics of hyperspectral data, namely, low rankness in the spectral domain and high correlation in the spatial domain. The Gaussian mixture model enables us to make a good estimation of Gaussian noise intensity and the locations of sparse noise. The proposed method takes advantage of the low rankness using subspace representation and the spatial correlation of HSIs by adding a powerful deep image prior, which is extracted from a neural denoising network. An exhaustive array of experiments and comparisons with state-of-the-art denoisers was carried out. The experimental results show significant improvement in both synthetic and real datasets. A MATLAB demo of this work is available at https://github.com/LinaZhuang for the sake of reproducibility.