Is there any work on defining something analogous to a generalized function for functions whose domain is a Hilbert or Banach space? Is there an extension of the notion of Frechet/Hadamard/Gateaux derivatives that somehow resembles the theory of distributions? I am hoping to take something like a Taylor expansion of a nondifferentiable function from a Hilbert space to the real line.

There is a substantial literature on spaces of distributions on infinite dimensional spaces, in particular function spaces, so much so that there is a subsection of the AMS classification scheme devoted to it (35R15). This might be relevant for your question. Related subjects can be found in 46G. This work is mainly motivated by questions in quantum theory, especially quantum field theory. A possible starting point would be the work of Y.M. Berezanskii, e.g., his monograph "Selfadjoint operators in spaces of functions in infinitely many dimensiona". 


Yes there is a notion of generalised function in infinite dimensions, see http://www.encyclopediaofmath.org/index.php/White_noise_analysis and the references contained therein. These functions are called "Hida distributions". 

