It is pretty much unprecedented for a U.S. state or a U.S. local government to have an office outside its territory at which one can conduct ordinary state or local government business with a state or city, outside very trivial examples of a city that may have an office in a building that partially within the city but extends across a municipal boundary, which may locate a few of its offices across the territorial line of that government (e.g. a DMV office in a mall in an office unit that crosses a county or state line).
Of course, lots of state and local governments do do business by telephone, mail, fax, wire transfers, credit card payment systems, interstate banking systems, and the Internet. And, this can make it possible to do business with these state and local governments from outside their boundaries.
But, there are other forms of extraterritorial operations by subnational governments that deserve mention as they implicated similar concerns and concepts.
Interaction, Cooperation and Coordination Without Service Provision
It isn't uncommon for state and local governments to have interactions with foreign governments.
For example, many U.S. cities have "sister city" relationships with a number of foreign cities that are similar to them in some respect, and for the sister cities to exchange delegations to visit and confer with each other.
Domestically, many U.S. local governments are part of organizations such as a "Regional Council of Local Governments" which functions as a forum in which local governments can negotiate compacts with each other and coordinate their policies for their common benefit. But the RCOG rarely provides services particular to a single member government to the general public. It is sort of like a state and local government version of the United Nations.
Similarly, state and local governments not infrequently impose sanctions on foreign governments in exercise of their non-regulatory powers to spend their own funds as they wish.
Extra-territorial facilities specific to the location of the facility
A few local governments, like the City and County of Denver, operate assets outside its own boundaries, such as a ski resort and a buffalo herd, which are in the nearby mountains in different counties.
Similarly, it isn't uncommon for airports owned by municipal governments are located outside the municipalities territory, or for local governments to operate transit services that bring someone from outside their boundaries to their boundaries (e.g. a suburban municipality that runs a shuttle service that takes people from an airport or seaport outside its boundaries to the suburban municipality).
Cross-Border Governmental Cooperation And Contract Service Provision
One other parallel phenomena that exists in a handful of places, but is also not strictly on point, is for parallel local governments that function as a unit to operate on both sides of a state or international boundary.
For example, College Corner is a town that is partially in Butler County, Ohio, and partially in a parallel town of the same name across the state line in Indiana. There is a College Corner school district on each side of the state line and the two school districts share a single set of students and a single primary building that straddles the state line, and has negotiated public funding appropriations from each state that are pooled in some sort of joint operating agreement. This example is particularly odd because Ohio and Indiana are in different time zones, so it can be, for example, 10 a.m. on one side of the school building, and 11 a.m. on the other, at exactly the same time.
Similarly, it isn't uncommon for one government to provide governmental services for another nearby government outside its territorial boundaries on a contract basis. For example, the City and County of Denver's fire department provides fire department services to the tiny adjacent city of Glendale, Colorado which has no fire department of its own, on a contract basis, even though this is outside its municipal boundaries.
Another service often provided across jurisdictional lines on a contract basis is prison incarceration in which one government that needs a place to house incarcerated people in its criminal justice system houses them in another jurisdiction's jails or prisons in exchange for payments to the government providing the service on a contract basis, especially, in cases of specialty prisons like psychiatric facilities, or maximum security facilities, or facilities for juvenile delinquent girls.
Also, in the law enforcement and wildfire protection service areas, there are often "mutual support agreements" in which government agree to cross territorial boundaries to provide services and assist a primary law enforcement or wildfire protection department of a territory, in the event of an extraordinary event that requires extra personnel, according to pre-negotiated terms and conditions.
Likewise, states and foreign countries and tribal governments sometimes enter into reciprocity agreements with other states and foreign countries, and tribal governments, to treat residents of other compacting entities as their own for purposes of providing higher educational services.
So, for example, maybe the Sioux Nation, or American Samoa, or Wyoming, or the Yukon in Canada, doesn't have a medical school that trains neurosurgeons. These governments might enter into an agreement with Colorado to treat their residents as Colorado citizens for purposes of medical school admissions and funding, while Colorado students who want to major in oceanography in college, might be able to receive similar treatment in American Samoa.
Autonomous Local Government Functions v. Administration For Higher Level Governments
Another factor to consider in evaluating the question is the nature of state and local governments in different countries. In the U.S., the federal government and state governments are legally considered autonomous parallel sovereigns that mostly are limited in how they can interfere with each other, that are subject to the same rules of the road. State governments and home rule cities in the U.S. have a similar relationship.
But, in the U.S., school districts and county governments are far less autonomous from the state governments that created them, even though they have their own elected boards or elected officials. Locally elected district attorneys enforce state criminal laws and send convicted felons to state prisons, for example. These local governments implement state law through territorially based local geographic divisions that typically cover 100% of the territory of the state. There are some federal programs in the U.S., like Medicaid and the junk fax law, that operate this way as well.
The relationship between the national government and state governments in many subject matters areas of the government of Germany is similar. Rather than having a large central government bureaucracy, lots of federal government and federal law governed programs are administered by state governments.
In circumstances where regional, state and local governments act of territorial local offices of a higher level government, rather than as truly autonomous local governments, the notion that an extraterritorial office carrying out a state or local government function breaks down conceptually, because most of the functions carrying out are actually functions of higher level governments that are merely administered in many local offices that are run by local elected officials.