Although it was never intended as a standard battle rifle, I think the M1 Carbine would qualify as a very early assault rifle. It did not have a full auto feature but that was added later.
In my opinion, it was better without it.
The M1 Carbine fires a pistol cartridge. You would never assualt with it. It is designed for CQB support of the M1 Garand, the same as the German MP40.
The original ar-10 was far superior to the m16, and should have been the main battle rifle, but politics won out.
There are lots of arguments about the M-1 Carbine and the MP-43.
Certainly the M-1 carbine had been captured by the Germans in 1942 in its self loading configuration.
Certainly the M-1 Carbine was issued in 1944 in a fully automatic version with a 30 round magazine.
Oddly, both had the same cartridge length: 33 mm. Heavier rounds with the same velocity have longer range. US Carbine round gave up both weight and velocity to the German 7.92x33 round.
The German M43 round (125 grain bullet, 7.92x33K) was bottlenecked and gave slightly higher velocity ~2,250 fps per Wikipedia. The 7.62x33 US round was lighter and slower velocity (110 grain, 1,990 fps).
By contrast the Soviet M-43 round (7.62x39) was slightly lighter, and faster (123 grain bullet 2,421 fps).
I have on my wish list a broom handle mauser that will fire the M-1 Carbine round. The original broom handle Mauser couldn’t take the recoil, but springs are better these days, and I hope some designer would take a shot at it.
(The original .30 Mauser had an 86 grain bullet and 1450 fps round) The broom handle Mauser was designed for roughly the same thing as the M-1 Carbine: An officer, cavalry, or artillery crew auxiliary weapon that wouldn’t get in the way, but could be used in emergencies at short range.