Has there ever been a war between two service branches in the same country?

Alexandre Dumas wrote the Three Musketeers which details fighting between the army and the Kings Musketeers?

Also more of a battle, but how about Japan 1945 the Kyūjō incident? It was an abortive military coup d'état at the end of the Second World War. 14–15 August 1945, just before the announcement of Japan's surrender to the Allies. Organized by the Staff Office of the Ministry of War of Japan and participation from the Imperial Guard of Japan designed to stop the move to surrender.

The officers killed Lieutenant General Takeshi Mori of the First Imperial Guards Division and attempted to counterfeit an order to the effect of occupying the Tokyo Imperial Palace. They tried to place the Emperor under house arrest, using the 2nd Brigade Imperial Guard Infantry. They failed because they couldn't persuade the remaining army to oust the Imperial House of Japan. They coup leaders ultimately committed suicide. As a result, the communiqué of the intent for a Japanese surrender continued as planned.

The Nazi's SS vs the Nazi's SA ( brownshirts). The Brown Shirts (SA) origins in WWI predated the Nazi's; however, they played a major role in the Nazi's coming to power. The Brown Shirts were the Nazi's enforcers. After the Nazi's achieved national power, Nazi's crushed the SA as a way to both consolidate more power under Hitler, and appease the German Military which was opposed to what they saw could have been a rival in the SA.

At the end of 1933 the SA had 3 million members. The German army at the time was legally limited to 100,000 men. As the Nazi's came to power one of the goals of the SA was to absorb the German Army into it's ranks and rebrand itself the "People's Army" of Germany. It cannot be overstated how little the German army(General Werner von Blomberg, the Minister of Defense, and General Walther von Reichenau, the chief of the Reichswehr's Ministerial Department) through of this idea. So the German Army was all about getting rid of them. Purging the SA was made a condition of the German Army's cooperation with the Nazi's in the early 1930's.

Inside the Nazi party, Hermann Göring and Heinrich Himmler, started plotting against the SA and it's leader Ernst Röhm. Himmler the head of the rival SS, decided the SS could never come into it's own if it was overshadowed by the SA. I do not know why Goring was an early detractor of the SA, perhaps it was his German Army background, Goring being a war hero from the first war. Matters came to a head in June 1934 when President von Hindenburg informed Hitler that if a move to curb the SA was not forthcoming, then he would dissolve Hitler's Government and declare martial law. Hitler would dismantle the SA and replace it with a newly empowered SS which would owe him it's loyalty. This resulted in the "Night of the Long Knives", June 30 1934. On 28 June, Hitler phoned Röhm and asked him to gather all the SA leaders at Bad Wiessee on 30 June for a conference. Röhm agreed, apparently unsuspecting.

The Night of the Long Knives began at dawn of that day, Hitler flew to Munich and drove to Bad Wiessee, where he personally arrested Röhm and the other SA leaders, who were all consigned to Stadelheim Prison in Munich. From 30 June to 2 July 1934 the entire leadership of the SA was purged (most of them shot), along with many other political adversaries of the Nazis.

There was the Battle of Castle Itter, at the end of WW2, where Wehrmacht and a small number of US forces joined together to fight off a SS division attempting to recapture the Castle Itter. The castle was a prison for high ranking French political prisoners. Presumably, the SS intended to capture the castle to execute the prisoners.

Relief arrived after prisoner and peacetime tennis star Jean Borotra had run on foot from the castle to the main US forces some miles away, to alert them to the precarious position of their advance unit.

Yes, it happened. Cavalry vs infantry. With some casualties. But it was an accident: the Battle of Karánsebes.

The definition of war

a state of usually open and declared armed hostile conflict between states or nations

War is between states.

If you have two different service branches fighting, then they are pretty much by definition not from the same country. Might be a mutiny or a coup. Might be a civil war, but in all of those cases, the two forces are serving - or attempting to serve - different states.

I'm not sure if it counts as a "war", but the Battle for Castle Itter on May 5, 1945 was certainly a battle between two service branches of the same country - the Wehrmacht and the Waffen SS. A unique feature of this battle was that the Wehrmacht were allied with a unit from the US Army.

Schloss Itter is a medieval castle in the Austrian North Tyrol village of Itter. During the Second World War it became a sub-unit of the Dachau concentration camp used for VIP prisoners, prominent politicians and military figures that the Nazis wanted to use as bargaining chips. Prisoners there included two former prime ministers of France, Edouard Daladier and Paul Reynaud, as well as Marie-Agnes Cailliau, the elder sister of General Charles de Gaulle.

In May 1945, the German guards at Schloss Itter fled, but the prisoners were trapped in the castle by roaming units of Waffen SS and Gestapo. The prisoners managed to contact a German major, Josef (Sepp) Gangl who had become opposed to the Nazis and was collaborating with the Austrian resistance.

At that time, Gangl only had a couple of dozen loyal troops with him. Under a flag of truce, he made contact with the nearest American unit - from the 23rd Tank Battalion of the US 12th Armoured Division, commanded by Captain Jack Lee.

Lee offered to lead a rescue mission to the castle.

They reached the castle, but at dawn on 5 May, they were attacked by a large force of Waffen SS. The SS succeeded in blowing up the American Sherman tank protecting the castle gate, but were unable to storm the castle. The defending forces held out until a relief force from the 142nd Infantry Regiment arrived, encircling and capturing about 100 Waffen SS troops.

The combined US-German force only suffered a single casualty - Major Gangl was killed by an SS sniper.

The battle is often remembered as the day the US Army joined up with the Wehrmacht to fight the SS.

Addendum:

See also: Operation Cowboy - Wikipedia, which took place in the 28th of April 1945 and is one of two known incidents during the war in which Americans and Germans of the Wehrmacht fought side by side against the Waffen-SS.