Yes, though do not think of it as a job you apply for (only the Russians and East Germans ever really did have actual official assassins working for them in that direct capacity). Over the period of the Cold War where this kind of thing really happened with anything even approaching regularity- and even then it was rare- Government security agencies would have names of people they could use if they needed something terminal done.
Sometimes Government agencies do try directly- we all know how hilarious the CIA's attempts to kill Castro were. Of course, the really successful ones you never hear about... but even then, the CIA nearly always got someone else to do the job for them, and these days assassinations are more likely to be done by bomb and missile than a poisoined cigar or umbrella tip (failed and very successful respectively).
In the underground war with the IRA, things tended to get comical also (the original FRU (our specially created security Agency for Nothern Ireland) security operation was basically a laundry firm, which would collect the washing of all suspected IRA members and search thge clothing for evidence against them. Seriously! It was a proper laundry and everything). But also very harsh, as the British security agencies (or at least some elements of them) would simply identify Republican targets to Unionists, knowing full well the Unionists would then kill the target for them. The British security personel felt justified doing that seeing as the IRA would just shoot them dead in the street, of course, but then that was a HORRIBLE conflict.
So all in all... Governments kill, but they don't really hand out licenses to kill- they go to more obscure lengths to do it.