We show convergence rates results for Banach space regularization in the case of oversmoothing, i.e. if the penalty term fails to be finite at the unknown solution. We present a flexible approach based on K-interpolation theory which provides more general and complete results than classical variational regularization theory based on various types of source conditions for true solutions contained in the penalty's domain. In particular, we prove order optimal convergence rates for bounded variation regularization. Moreover, we show a result for sparsity promoting wavelet regularization and demonstrate in numerical simulations for a parameter identification problem in a differential equation that our theoretical results correctly predict rates of convergence for piecewise smooth unknown coefficients.