There's no need to capitalize, it should be just 'ancient.'
It's a descriptive adjective, not part of the proper name. Similarly with most other chronological descriptions: early modern English, ancient Zoroastrianism, medieval Scholasticism, pre-war Germany, etc.
Since this just came up a day or so ago when someone mentioned ancient Christianity, I'm happy to be able to quote the Chicago Manual (15th ed., pp. 340-42):
8.78 Descriptive designations. A descriptive designation of a period is usually lowercased, except for proper names. ...
the antebellum period
the baroque period
the colonial period
a golden age
the Hellenistic period
the romantic period (see also 8.85)
the Shang dynasty
the Victorian era
8.79 Traditional names. Some names of periods are capitalized, either by tradition or to avoid ambiguity. ...
the Age of Reason
the Augustan Age
the Common Era
the Dark Ages
the Gay Nineties
the Gilded Age
the Grand Siècle
the High Middle Ages (but the late Middle Ages)
the High Renaissance
the Jazz Age
the Mauve Decade
the Middle Ages (but the medieval era)
the Old Kingdom (ancient Egypt)
the Old Regime (but the ancien régime)
the Progressive Era
the Roaring Twenties
romanticism, romantic (sometimes capitalized to avoid ambiguity)
That said, yes, people within a particular area have a strong tendency to capitalize familiar designations because they loom so large in their view that they begin to feel like proper names. The over-capitalization disease afflicts everyone from Business Managers to Universities [sic]. But we should all resist that temptation. (-;