We develop a theory for functional independent component analysis in an infinite-dimensional framework using Sobolev spaces that accommodate smoother functions. The notion of penalized kurtosis is introduced motivated by Silverman's method for smoothing principal components. This approach allows for a classical definition of independent components obtained via projection onto the eigenfunctions of a smoothed kurtosis operator mapping a whitened functional random variable. We discuss the theoretical properties of this operator in relation to a generalized Fisher discriminant function and the relationship it entails with the Feldman-Hájek dichotomy for Gaussian measures, both of which are critical to the principles of functional classification. The proposed estimators are a particularly competitive alternative in binary classification of functional data and can eventually achieve the so-called near-perfect classification, which is a genuine phenomenon of high-dimensional data. Our methods are illustrated through simulations, various real datasets, and used to model electroencephalographic biomarkers for the diagnosis of depressive disorder.