If a country has the comparative advantage in producing a product, then must that country also have the absolute advantage in producing that product?

No! That's the really fascinating thing about comparative advantage; you can have comparative advantage in something without having absolute advantage in anything---you can be worse than everyone else at doing literally everything, and it still makes sense for you to do some things because you're less worse at that than you are at everything else.

For a surprisingly realistic example, let's compare the United States and Vietnam. Furthermore for simplicity let's pretend there are only two goods in the world, shoes (which you can think of as standing in for low-tech textiles) and cars (which you can think of standing in for high-tech capital-intensive manufacturing). Nothing else, just shoes and cars.

Now suppose that the United States has the following production possibility frontier: They can produce up to 100 million cars, or up to 10 billion pairs of shoes. Again for simplicity, let's assume their PPF is linear, so they have a constant rate of substitution where they can sacrifice making 1 car in order to make 100 pairs of shoes.

Vietnam, by contrast, could only produce up to 1 million cars, but can still produce up to 1 billion shoes. By sacrificing making 1 car, they can build 1000 pairs of shoes---thus, they clearly have a comparative advantage in making shoes.

Yet, notice, the US still has an absolute advantage in everything; they can make 10 times as many shoes or 100 times as many cars.

Depending on how we value cars versus shoes, it is very likely advantageous for both countries if Vietnam makes most or all of the shoes, while the US makes all the cars, and then they trade.

For example, if the US wants 500 million pairs of shoes, they could do that, but they'd only be able to make 95 million cars instead of 100 million. Meanwhile if Vietnam makes 1 million cars, they can't make any shoes at all.

Whereas, if the US makes 100 million cars and sells 2 million of them to Vietnam in exchange for 500 million shoes, Vietnam gets 2 million cars---more than they could ever produce on their own, while still getting to keep 500 million shoes. Meanwhile the US gets 98 million cars and also the 500 million shoes they wanted. Both countries are better off as a result of the trade, because Vietnam specialized in their comparative advantage of shoes.