The answer, unfortunately, is yes - if your income is high enough.
Your earned income is taxable, so you do have to pay income tax like everyone else if you earn over the HMRC's 'personal allowance'. For the latest information on Income Tax Rates and Personal Allowances, please see Income tax rates & Personal Allowances Therefore, if your earned income is under that figure, you will not need to pay tax.
You need to make sure you are paying the correct amount of tax and that your employer knows that you are a full-time student so that your earnings are recorded on the correct tax code. If you have been paying tax on an incorrect tax code and you earn only a small amount over the financial year, you may be entitled to a Income Tax Refund