Does Wiki give a source? A date?
Sorry, I haven't got the time to check it out at the mo'.
Looking at the French Wikipedia (and seems like other sites agree), the arms there were granted by Henry III in 1560, shortly after retaking the city from the English.
Silly question from me, but do we know what the English policy was around assigning municipal heraldry at the time? I'm wondering if potentially it might just be a case that the city would use the Royal Arms or the arms of whoever the Captain of Calais is at a given time?