Our County Sheriffs can do this, although they usually have some pre-organized auxiliaries who can be called up if a manhunt or search-and-rescue is in order.
That “Magna Carta” form of Posse Comitatus remains a part of New Zealand law, as we brought most of our laws from the UK when we became a country. It has not been repealed yet.
This means that, even though we have a national New Zealand Police force, *theoretically* any mayor should be able to establish his/her own policing capability — even a casual one — if he/she deems fit. Hasn’t been tested in court yet, but that is certainly how our laws seem to read.
So *theoretically* parking wardens might be called upon by the mayor to catch a murder suspect... or possibly even unwarranted citizens could do the same.
It’s an exciting concept that warrants further research. Which is precisely what I am doing, seeing how I lead a civilian volunteer organization in NZ that focuses on fighting crime.