The Germans were developing guided antiship missiles for launch from airships in WW1. The payload was a torpedo.
The UK developed a prop powered SSM called the Larynx (?) that saw combat use in Mesopotamia in the '20s. Some versions may have used radio guidance.
There were also WW1 projects in the US and UK for flying bombs which Gunston describes as SSMs. Thousands were produced in anticipation of the planned 1919 offensive. They had no guidance other than an autopilot and timer.
Bill Gunston's
Illustrated Encyclopedia of the Worlds Rockets and Missiles has entries on these and other early missiles.
Regards the missile in question:
The USAAC began developing an AAM in 1937 called Gorgon. It went through several designs switching to turbojet and then back to rocket power. The actual production run of the AAM version around 25 and was completed by 1943 . According to Gunston it ultimately spawned families of experimental SSMs and SAMs that were at least test beds into the late 40s. Robert Truax was involved in the development towards the end so that might be a good place to start looking for info.
Update: Or you could just click here....
http://www.designation-systems.net/dusrm/app1/asm-n-5.html
Note that some of the info there contradicts mine, which comes via Gunston. Designation Systems is much more recent and has lots of photo's so I'd go with them.