'Deep State' is Obama's legacy
The term in the US refers to those burrowed into the bureaucracy that are hostile to a duly elected President and his administration. All the leaks to the media since Trump was elected seem to validate the observation. You get the impression that Obama left land mines as he retreated from office and his people are supporting the deep state effort to sabotage the Trump administration.
The term is most often used to describe countries with authoritarian elements, like Egypt, Turkey and Pakistan, not the United States.