'Big Data' classification is hindered by the large volume of often high-dimensional data, missing or absent features and, in streaming operation, the need for real-time processing. This paper aims at learning a kernelized support-vector-machine (SVM) classifier from (generally nonlinearly separable) large-scale incomplete data 'on the fly.' Leveraging the low-rank attribute of the (even incomplete) data matrix, a novel online algorithm is developed for tracking the latent linear subspace jointly with the nonlinear classifier. Tailored for big data applications, dimensionality reduction based on the learned subspace is carried out online, while at the same time seeking the classifier in the reduced dimension. Performance analysis along with preliminary tests corroborate the effectiveness of the novel approach.