Which country first developed ballistic missiles?

China invented gunpowder powered rockets in the 13th century during the Song Dynasty which were called fire arrows 火箭, but rockets didn't become practical until the 20th century. Dr Robert H. Goddard a physicist from Worcester, Massachusetts, in the United States started experimenting with different types of rockets and fuels. He developed a liquid fueled rocket which he successfully launched on March 16 1926. The Germans, however, were the ones who made the first ballistic missile during World War II, it was the V2 rocket.

The V2 ( Vergeltungswaffe) rocket was not only the first true ballistic missile, it was also the first man-made object to leave the atmosphere and enter space. It was developed by Dr. Wernher Von Braun, who expanded upon Goddard's work for the German Wehrmacht in the early 1940s. The V2 used a liquid ethanol fuel and oxygen system and launched a 2,000 pound warhead to an altitude of 264,000 feet (50 miles or 80 Kilometers) in a 120 mile ballistic trajectory using both propulsion and gravity to guide it toward it's intended target. The early rockets used a simple analogue computer for guidance while the later models used a radio guidance system. More than 3,000 V2 Missiles were launched before the end of World War II.

Dr. Von Braun immigrated to the United States and was instrumental in developing not only American ballistic missiles, but also the multi stage Mercury, Atlas, and Saturn V Rockets that launched American Astronauts to the moon.