  In the present work, we discuss variational regularization for ill-posed
nonlinear problems with focus on an oversmoothing penalty term. This means in
our model that the searched-for solution of the considered nonlinear operator
equation does not belong to the domain of definition of the penalty functional.
In the past years, such variational regularization has been investigated
comprehensively in Hilbert scales. Our present study tries to continue and to
extend those investigations to Banach scales. This new study includes
convergence rates results for a priori choices of the regularization parameter,
both for H\"older-type smoothness and low order-type smoothness. The necessary
tools for low order smoothness in the Banach space setting are provided.
