  We extend the theory of operator-valued frames (resp. bases), hence the
theory of frames (resp. bases), for Hilbert spaces and Hilbert C*-modules, in
two folds. This extension leads us to develop the theory of operator-valued
frames (resp. bases) for Banach spaces. We give a characterization for the
operator-valued frames indexed by a group-like unitary system. This answers an
open question asked in the paper titled "Operator-valued frames" by Kaftal,
Larson, and Zhang in \textit{Trans. Amer. Math. Soc.} (2009). We study
stability of the extension. We also extend Riesz-Fischer theorem, Bessel's
inequality, variation formula, dimension formula, and trace formula. Further,
notions of p-orthogonality, p-orthonormality and Riesz p-bases have been
developed in Banach spaces and Paley-Wiener theorem has also been generalized.
We derive `4-inequality,' `4-parallelogram law,' and `4-projection theorem.'
