@Stéphane ********
If IO accepts your application and stamps you, you will have 7days to leave from the application day.
Usually, they wouldn't do it and just turn you away without accepting the documents, your stamp is still valid until the last day in March. But to make sure, you can 'ask' them if you can apply for it verbally first. Your current stamp is not touched and still is valid as it is, even if they say no.