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?
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:
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:
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.
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.