Do you ever say it? Yes, sometimes
Is it an old form that no one uses anymore? No, it isn't. You will hear people say this.
Can I come across it in books? Yes, but not only in books.
In British English, we sometimes use just 'haven't' - as an alternative to 'haven't got' - in a few specific constructions. The phrase "I haven't a clue" is one good example. Another time we sometimes use 'haven't' on its own is when it's followed by 'any' 'much' or 'many', as in "I haven't any money" or "We haven't much time".
Those are the exceptions to the rule, however. In most situations, we use either 'I don't have..' or 'I haven't got..' in just the same way as everyone else in the world does.
There are a lot of myths surrounding the use of 'haven't' which you're asking about. At one end of the scale, you have the ancient textbooks, unfortunately still in use in Russia, which would have you believe that "Have you sisters?" "No, I haven't sisters" is normal English. (It isn't . Nobody has had that conversation in that way since about 1930).
You will also meet American English speakers who'll tell you that they don't speak like that (true) but that the British people do ( untrue).
Then at the other end of the scale, you'll meet people who'll tell you that the form 'haven't' as the negative of 'have' is archaic and only exists in books (not true either).
We do still use it - but only sometimes and in certain phrases. Is this something that you need to learn about? No, it isn't. There are far more important aspects of English for you to concentrate on.
'Don't have' is always right.
'Haven't got' is sometimes, but not always right.
'Haven't' is usually wrong or very unnatural. You can forget about it.