The first space-controlled theremin was constructed in 1920. Both A. Smirnov (Sound in Z) and A. Glinsky (Ether Music and Espionage) agree Lev built his prototype pitch-only etherphone in Petrograd in 1920, demonstrated an instrument to his supervisor in October 1920, gave his first public "concert" in November 1920, and filed a Russian patent in June 1921. German and American patents followed in 1924 and 1928.

However, Lev Termen himself maintained that his creative process for an electronic instrument controlled without contact (and by his own reckoning, the "moment of invention" itself) occurred while he was working as a radio engineer in the Bolshevik Army ~1918-9:

At the beginning of 1917 I was assigned to a military school for six months, and then to a higher military engineering school, which I graduated from as a lieutenant, and was assigned to a radio engineering battalion in Petrograd. In 1918 ... for the first time I did things that concerned music (two university students helped me out) .... I made a strong transmitter-receiver, and suddenly there was too much feedback, a strong sound interaction. And it turned out that when the capacity changes at a distance of the moving hand, the pitch of the sound also changes. I immediately tried to play this sound with my hand. This was the moment of invention. ("Это и был момент изобретения.") Then [in 1919] I moved to Petrograd and headed the transmitter in Tsarskoe Selo"

- Interview with Lev Termen conducted by E. Petrushanka, published 1995, sent by Andrey Smirnov.

The basic principles of wirelessly heterodyning space-controlled instruments seem to have been noticed independently by many radio engineers at the time of the First World War. The principle was noticed by Maurice Martenot, by Lee de Forest, and also by British radio telegraphists in France, as reported to the British press by Jasper Maskelyne. None of these engineers constructed a viable space-controlled instrument. To conclude: we might say that the “theremin principle” was independently recognised by several individuals in the late 1910s. Lev Termen discovered this principle himself sometime in 1918/19. The first true theremin instrument employing this technology was built and demonstrated in 1920.

A note on nomenclature: for the first few years of its existence, the instrument was known under different names including “the apparatus”, “etherphon(e)”, and “radiophon(e)”. The instrument later gained the monikers “termenvox” (“voice of theremin”), “theremin(o)vox”, and “theremin”. In 1928, Leon patented his instrument in the United States as the “theremin method and apparatus for the generation sounds”. In 1929, the Radio Corporation of America selected the names “Thereminvox” and “Theremin” as those under which to market its test run of commercially-manufactured instruments.

I've attached some photos: of Lev as a member of the reserve Electrotechnical Batallion in 1918 (AG); of the earliest documented instrument demonstrated by Lev in 1921 (AS); and of the laboratory in Petrograd where he was working at the time which include one of him standing proudly beside his mighty television apparatus. You can also make out some amplifying horns in the background.