Sometimes the FLUX can do that ... it's designed to "clean" the copper, and it does that by chemically attacking the copper oxide layer on the surface. On VERY thin wire the chemical action may be enough to dissolve not only the oxide but the underlying metal too.
Try a different brand of solder ... maybe try something like silver solder.
Solder dissolves copper, silver, aluminum (I think), and other metals at various rates. Usually it's not an issue - you just run fresh solder across a joint that is becoming difficult from having too much copper in it. It sounds like you have a very hot iron and very thin wire. A professional soldering iron with a thermostat should eliminate the problem. Silver alloy solder is less prone to dissolving metals but it won't be a complete cure for an iron that's too hot.
"Mr. B1ack"
** Pretty wild piece of speculation - seeing as ordinary solder flux is sometimes used to coat and protect the copper layer on PCBs. ** Nuts..... Phil
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Einstein said it: "it is all relative"; fine wire making it stand out).
-- http://aws.org/wj/supplement/WJ_1975_10_s370.pdf JF
One thing to try is to take some solder and add some fine copper wire to it - a propane torch or hot iron good for this - then let the solder eat at the copper until its appetite is lessened. Now use that to coat your iron with.
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