I see two problems with your sentence.
Problem #1: Without additional context, there is no way of knowing to what the "they" at the beginning of the sentences refers. What conveys? It is difficult to talk about the sentence without knowing this.
Problem #2: If you choose "their" in your "its/their" choice, then "they" and "their" will naturally be interpreted as references to the same thing, even though it is your intention that "their" be a reference to "people".
That said, and knowing that you intend "its/their" to be a reference to "people", "it" and "their" would both be acceptable choices. You are telling me that "people" means "nation" but there is nothing in the sentence that emphasizes that choice. "People" can also be the plural of "person". If it your intention to use "people" to mean "nation", then you need to say something like this:
"the cultural life of THIS people, its way of thinking, and history"