"Ancient Arabic"? Probably you are right that they meant classical Arabic. Ancient Arabic is not a very comprehendable term. Nabatean is considered the father of classicial Arabic used till today.
If they are refering to classical Arabic as ancient, it is definitely a wrong word, since most Arabic dialects now are not necessary a development of Classical Arabic. Classical Arabic is the Quranic Arabic. Quran was revealed to the prophet mainly in the dialect of Quraish tribe, the most powerful and dominant tribe of that time in the area. Due to Islam spread and the Quran too, the Quraishian dialect became spread and was recognized as the official state dialect, and later, the classical dialect of Arabic. Today, the classical Arabic language is used with injected modern phrases and terminologies and redefined as the the Modern Standardized Arabic (MSA). The dialects in the Arab words are due to a derivative of the classical Arabic (Quraishian dialect), or another Arabic dialect that was carried by waves of immigrants Arabs to the conquested regions. Most Arabs today are taught classical Arabic in schools and it is widely used on TV, newspaper, and books along with publications.
Sorry for the long explanation!