I know South Korea and North Korea are still in a state of war. Similarly for Israel and Lebanon. And it seems Russia and Japan are also, technically, in a state of war. The last, if true, is surprising (as it is not something you would think every day). I wonder if there are other armed conflicts between recognised countries that I am not aware of? (so Israel-Palestine is another example, depending on the legal view of Palestine as a country)
Correlates of War and Inter-State Wars
The best data set I know of for war-related research is the Correlates of War (CoW) project. CoW has been operating at the University of Michigan since the 1960s and has been used in many professional (academic) publications.
They categorize wars into four types based on who is fighting. You are describing inter-state wars (wars fought between different states). Their data was last updated in 2007. At that time, there were no on-going inter-state wars. The most recently completed war is the United States invasion of Iraq in 2003.
CoW publishes a guide describing how the count and categorize wars. One important caveat is that they don't count anything with less than 1,000 battle-related fatalities among all participants. So it's conceivable that small wars are slipping by.
This is not an entirely correct answer but sort of interesting. I have sometimes read that the Second Seminole War of 1835-1842 is technically still ongoing since there never was a peace treaty signed.
Note that technically the Seminole people are not a country, being considered to be a domestic dependent nation within the USA. I don't believe that the US Congress ever declared war on the Seminoles but merely appropriated funding for the war effort.
As the war wound down various Seminole groups agreed to leave Florida and be transported to The Indian Territory. There never was anything like a peace treaty with all the Seminoles. In 1842 Colonel Worth received permission to declare the war over, and did so on August 14, 1842.
The Third Seminole War ended the same way, with many Seminoles captured or surrendered and being shipped to The Indian Territory, and Colonel Loomis declaring the war over May 8, 1858.
The Second Seminole War can be considered legally similar to the American Civil War, with a group of US subjects and/or citizens revolting against the government. There was never any peace treaty with the CSA. Instead President Andrew Johnson proclaimed that that the insurrection was ended in all the south except Texas on April 2, 1866, and a total end to the war on August 20, 1866:
"And I do further proclaim that the said insurrection is at an end and that peace, order, tranquility, and civil authority now exists in and throughout the whole of the United States of America."
So if anyone claims that the Second Seminole War is still ongoing, a sort of "frozen war", would they also claim that the US Civil War was also a sort of "frozen war", or would they claim that they are different?
They could claim there is an important difference between a proclamation of the war's end by a local military official and by the President, for example. Or they could claim that the federal government of the Confederacy no longer exists and that Seminole tribal governments do still exist, making the Seminoles more capable of waging war than the Confederacy is.
So I suppose that there is room to question whether the Second Seminole War has officially ended in a peace or if the fighting has merely slowed down and stopped for over a century and a half?