According to the "

no hair theorem", there are 3 parameters that can be detected outside a black hole: mass, electric charge, and angular momentum.

So, yes, an electric field (and a gravitational field) still extend to infinity, even from behind the event horizon of a black hole (but of course, getting weaker with distance, according to the inverse square law).

As I understand it, the angular momentum affects the shape of the event horizon, and can be detected by frame dragging outside the event horizon. But these effects are much more local to the black hole, and so cannot be detected at such great distances as the electric and gravitational fields.