Direct proof of use
The documented AIM-120 AMRAAM use in the 2011 Syrian Civil War is the 18 June 2017 shootdown of a Syrian government Su-22 near Ja'Din, south of Tabqah. A U.S. Navy statement said pro-Syrian regime forces attacked the Syrian Democratic Forces-held town of Ja'Din and that, at 6:43 p.m. Syria time, a Syrian regime Su-22 dropped bombs near SDF fighters before a U.S. F/A-18E Super Hornet shot it down in collective self-defense of coalition-partnered forces.
The Navy statement identifies the shooter aircraft and the defensive context, while later pilot-account reporting identifies the second air-to-air missile as the AIM-120 AMRAAM. Navy Times reported that Lt. Cmdr. Michael Tremel first attempted an AIM-9X shot, then turned to the AIM-120, which destroyed the Su-22.
Sources: U.S. Navy Coalition Defends Partner Forces, Navy Times Tremel Account
Timeline
The engagement followed a same-day ground clash around Ja'Din. Coalition aircraft first conducted a show of force after pro-Syrian regime forces attacked SDF positions, and the coalition also used the Russia deconfliction line before the later Su-22 bombing run.
After the Su-22 dropped ordnance near SDF fighters, Tremel warned and engaged the aircraft during a short air-to-air sequence. The War Zone's account of the Tailhook pilot discussion states that the AIM-9X shot did not bring down the aircraft and that the AIM-120 then hit the Fitter as it flew away from the attack.
Sources: U.S. Navy Coalition Defends Partner Forces, The War Zone Syrian Su-22 Account
Narrative
The AMRAAM appears in this conflict as a fighter-launched air-defense weapon used by a U.S. Navy Super Hornet protecting coalition-partnered ground forces, not as a transferred Syrian battlefield munition. The operator was the U.S. Navy within the U.S.-led coalition's anti-ISIS air campaign, and the immediate protected force was the Syrian Democratic Forces south of Tabqah.
The event was also notable as a rare U.S. air-to-air engagement over Syria. Military.com reported from Tremel's award coverage that the Syrian attack jet ignored warnings, dropped ordnance near friendly forces, and was destroyed after Tremel fired two air-to-air missiles, with the second identified as a radar-guided AIM-120 AMRAAM.
Sources: U.S. Navy Coalition Defends Partner Forces, Military.com Tremel DFC