The theory DCF${}_p$A exists for $p>0$

We prove that the (elementary) class of differential-difference fields in characteristic $p>0$ admits a model-companion. In the terminology of Chatzidakis-Pillay, this says that the class of differentially closed fields of characteristic $p$ equipped with a generic differential-automorphism is elementary; i.e., DCF${}_p$A exists. Along the way, we provide alternative first-order axiomatisations for DCF (differentially closed fields) and also for DCF${}_0$A.