  The purpose of this paper is to establish Picard-Lindel\"{o}f theorem for
local uniqueness and existence results for first-order systems of nonlinear
delay dynamic equations. In the linear case, we extend our results to global
existence and uniqueness of solutions on the entire interval (allowed to be
unbounded above), and prove the variation of parameters formula for the unique
solution in terms of the principal solution. A simple example concerning the
ordinary case is also provided.
