  The existence of unique solutions is established for rough differential
equations (RDEs) with path-dependent coefficients and driven by c\`adl\`ag
rough paths. Moreover, it is shown that the associated solution map, also known
as It\^o-Lyons map, is locally Lipschitz continuous. These results are then
applied to various classes of rough differential equations, such as controlled
RDEs and RDEs with delay, as well as stochastic differential equations with
delay. To that end, a joint rough path is constructed for a c\`adl\`ag
martingale and its delayed version, that corresponds to stochastic It\^o
integration.
