A title in the IELTS Reading Section reads -
The Phoenicians: an almost forgotten people
It's IELTS and thus unlikely to make a mistake!
What am I missing? Or IELTS has gone wrong?
Nothing has gone wrong. I agree that we've all gotten used to the idea of thinking of the word people as the plural form of person, but, believe it or not, it can also be used as a countable noun in singular (a people) as well as plural (peoples) form. It basically means a body of persons living in the same country under one national government. I simply recommend thinking of the phrase a people (or peoples) as equivalent to the phrase a nation (or nations).
So, the title is absolutely fine because, as far as I know from ancient history, the Phoenicians indeed were a people. It was a nation living in what's now called the Middle East. Though, I must say the term nation, strictly speaking, does not really apply here very well because nation is a more modern political concept which historically was a much later invention compared to when the Phoenicians lived.
Long story short, people when used without an article just refers to a bunch of individuals (in fact, it can be millions of individuals). People when used with an article refers to a large group of individuals living together as a nation (they speak the same language, share the same customs and live on the same territory).
Here's a reference for you from the Collins English Dictionary:
A people is all the men, women, and children of a particular country or race.
the native peoples of Central and South America