  Quasi-invariant and pseudo-differentiable measures on a Banach space $X$ over
a non-Archimedean locally compact infinite field with a non-trivial valuation
are defined and constructed. Measures are considered with values in
non-Archimedean fields, for example, the field $\bf Q_p$ of $p$- adic numbers.
Theorems and criteria are formulated and proved about quasi-invariance and
pseudo-differentiability of measures relative to linear and non-linear
operators on $X$. Characteristic functionals of measures are studied. Moreover,
the non-Archimedean analogs of the Bochner-Kolmogorov and Minlos-Sazonov
theorems are investigated. Infinite products of measures are considered and the
analog of the Kakutani theorem is proved. Convergence of quasi-invariant and
pseudo-differentiable measures in the corresponding spaces of measures is
investigated.
