Monaco - Monte-Carlo is the only railway station in the Principality of Monaco
travel.stackexchange.com/users/4995/pnuts
Monaco (another European city-state) has only one train station .
Vatican City also has only one train station .
A tourist train serviced it weekly from Italy in 2015, but it is not clear if it is still running.
travel.stackexchange.com/users/47855/Robert Columbia
At time of writing -- but not for much longer -- the railway network in Laos consists of a single station, Thanaleng (near the capital Vientiane), which has passenger services only across the Mekong river to Nong Khai, Thailand: https://en.wikipedia.org/wiki/Thanaleng_railway_station
However, in 2021 the Vientiane-Boten link across the country to China is scheduled to open, and this will have a number of domestic stations: https://en.wikipedia.org/wiki/Vientiane%E2%80%93Boten_railway
travel.stackexchange.com/users/1893/lambshaanxy
Actually all trains in Liechtenstein are Austrian (since 1872) , with locomotive / conductor change in Buch, Switzerland.
In 1970, the passport control was done (for Switzerland) by a Liechtenstein police officer who got on the train in Buch and off at the first Austrian station Feldkirch without stopping.
On request my passport was stamped with a Liechtenstein stamp.
travel.stackexchange.com/users/95267/Mark Johnson
Since early 2018, there is passenger train service between Ethiopia and Djibouti .
It appears that all trains go from Addis Ababa to Djibouti City, so all trains in both countries are effectively international trains.
However, there are at least two passenger stops in each country, so it is also possible to take this train domestically; I'm not sure whether this counts under the criteria of the question.
Moreover, there are other passenger lines under construction in Ethiopia , and once these are completed and begin passenger service, there will be both domestic and international passenger trains in Ethiopia.
travel.stackexchange.com/users/31165/Michael Seifert
All trains in Liechtenstein are international (in fact, they all serve at least three countries).  Are there any other countries with regular passenger train services where all trains are international?
Countries with solely international trains
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<trains><international-travel><factoids>
travel.stackexchange.com/questions/84540
travel.stackexchange.com/users/2509/gerrit
Monaco and the Vatican City only have one station, and Laos has a single station (Thanaleng) that only has passenger services to Nong Khai, Thailand. Ethiopia and Djibouti have one station each, with trains only travelling from Addis Ababa to Djibouti. Monaco, Vatican City and Laos all have only one station that has a rail service to another country. Ethiopia and Djibouti have only one railway line that connects Addis Ababa to Djibouti City.
Monaco has only one station as does Vatican City. Laos has one railway line connecting to Thailand. Ethipia and Djibouti are connected by one railway line.