Thanks for your question.
I'm sorry, no. You cannot at the present time apply for indefinite leave to remain. There is actually no special provision for students to obtain indefinite leave to remain and so therefore such student who are eligible must apply for ILR on the basis of the long residence rules.
This means that you may apply for ILR after you have completed 10 years lawful residence in the UK and your time as a student may count towards meeting this criteria.
If you are still studying then you must continue to extend your visa as necessary or otherwise switch in to a further visa category to continue your stay here. If you switch to a workers visa then you can apply for a long residence after 3 years because both this three years and your previous ten count towards the 10 years required for ILR under long residency.
In consideration of your application for long residency the following you should also meet the following:-
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