Students don't get any particular tax free allowance for being a student.
Everyone gets £5,035 tax-free in the current tax year. However this is distributed throughout the year - so if you get paid weekly, you get that amount divided by 52 tax free each week, and any extra amount earnt has tax added. It works similarly if you are paid monthly.
There is no way of getting out of paying income tax. If you are a student and you work only during the holidays, then it is possible (by using an NT tax code and filling in a form for your employer) to get all of your tax free allowance "at once", because you estimate that you won't go over the tax free amount. If you fill this in then work more, you will obviously pay more tax for the rest of the year to make up for it.
If you work during term time then the above concession doesn't apply and you must pay tax as normal. In addition, if you used up part of your tax free amount during the holidays on an NT tax code, you will end up paying more tax than you would normally because the full amount you earn will be taxed rather than just the amount above the tax free amount.
If you pay tax normally for part of the year then stop working then you should receive a tax refund for the tax paid that actually forms part of your yearly tax free allowance.
National Insurance behaves seperately and everyone (including students) must pay this on every payslip with no exceptions or concessions like those mentioned above.
Essentially, there isn't a magic "no tax for students" law, and if you are working during term time you will normally pay pretty much the same as anyone else. As far as I'm aware this isn't a new thing, I don't think there has ever (in recent years, at least) been a lower tax or tax exemption for students.
(I've no idea where "you get 9k tax free" came from, that sounds like nonsense to me)