This paper proposes a novel framework called Gaussian process maximum likelihood for spatially adaptive classification of hyperspectral data. In hyperspectral images, spectral responses of land covers vary over space, and conventional classification algorithms that result in spatially invariant solutions are fundamentally limited. In the proposed framework, each band of a given class is modeled by a Gaussian random process indexed by spatial coordinates. These models are then used to characterize each land cover class at a given location by a multivariate Gaussian distribution with parameters adapted for that location. Experimental results show that the proposed method effectively captures the spatial variations of hyperspectral data, significantly outperforming a variety of other classification algorithms on three different hyperspectral data sets.