Do all states in a DFA must be connected?
I suspect a trick question in the material you are studying - there are no DFAs with an infinite number of states in the first place.

in fact, the whole point of D/NFAs is their finiteness. if you could have an inifinite number of (reachable) states in your NFA you'd actually be able to recognize any language; just take for every word in your language the trivial deterministic recognizer and connect a global start state with $\lambda$-transitions to the start state of each of these automata.