Yes, a dpda can have dead state
Basically a dpda is nothing but a dfa with 1 stack as a memory
Just like dfa is moves from 1 state to another and is state dependent with an addition of being dependent on symbol on the top of stack
The only way to accept a given string is either by landing at final state or by accepting empty stack symbol. There is no was to reject a string other than to be in a nonfinal state.
So to reject a string which will never be true u have to enter into a dead state