Support vector machines (SVMs) are well-established classifiers that are effectively deployed in an array of classification tasks. In this article, we consider extending classical SVMs with quantum kernels and applying them to satellite data analysis. The design and implementation of SVMs with quantum kernels (hybrid SVMs) are presented. Here, the pixels are mapped to the Hilbert space using a family of parameterized quantum feature maps (related to quantum kernels). The parameters are optimized to maximize the kernel-target alignment. The quantum kernels have been selected such that they enable the analysis of numerous relevant properties while being able to simulate them with classical computers on a real-life large-scale dataset. Specifically, we approach the problem of cloud detection in the multispectral satellite imagery, which is one of the pivotal steps in both on-the-ground and on-board satellite image analysis processing chains. The experiments performed over the benchmark Landsat-8 multispectral dataset revealed that the simulated hybrid SVM successfully classifies satellite images with accuracy comparable to the classical SVM with the radial basis function kernel for large datasets. Interestingly, for large datasets, the high accuracy was also observed for the simple quantum kernels, lacking quantum entanglement.