urgent advice

pls i am an international student here in the uk and just had a new born son who is less than 3 month to a british citizen. please can you advice me on what to do since i cant renew my student visa hence have to leave the country or can i stay in this country due to my son been a citizen here?. can i apply for a british passport for my son and if so how long can my child stay with me in my home country?
thanks for your usual cooperation

Could you not get a Fiance visa and marry the father? You would then be able to get indefinate leave to remain. As for your son, is it not possible for him to travel on your passport to your home country? Whilst it is no longer possible for children to travel on a parents UK passport, I know they can still do so on other nationalities.

If he was born in Britain to a British Father, then he is automatically a British Citizen. i think all you need to do is go and get the application form from the post office, fill it in, send it off with the fee and photos and you will get him a passport. You may also need a visa from your home country if he is travelling with you and cannot go on your passport.