A number of US states maintain representation abroad for trade and investment promotion purposes. This list is a couple of years out of date, but gives you an idea of the variety of such offices. Note that not everyone listed there may be an employee of the state sent abroad; Pennsylvania's "Authorized Trade Representatives" are apparently local contractors, and some are multi-state or even international consortia, such as the offices operated by the Conference of Great Lakes and St. Lawrence Governors and Premiers
For example, California (which would have the 5th largest economy in the world if it were its own country) maintains an overseas presence in China, Mexico, and Armenia (covering "California’s Trade and Investment Representative for Europe, MEA, and India") for trade and investment promotion.
Another sort of example is that some universities, including state-owned universities, have a foreign presence. For example, the University of California system owns La Casa de la Universidad de California en México to facilitate partnerships and exchanges with educational institutions in Mexico.
US states may have an office in Washington DC to help bend the ear of the federal government on matters of interest. The Governor of California has a representative in DC, as do a number of other states (see all those listed with a DC address). Puerto Rico has the Puerto Rico Federal Affairs Administration to represent its interests with the federal government. States can also hire lobbying firms to represent them in Washington DC even if they don't have a physical presence there. It's also not uncommon for large state university systems to have a federal Governmental Relations office in Washington DC, such as the University of California or Ohio State.