Will I face an overstay if I go to immigration one day late for my visa extension?

You will get extension as usual if you go next business day immediately after,. You will be stamped overstay but fine will be waived due to office closure.

If you want to avoid mentioning overstay in your passport, the only way is to get out the country before 23:59 10th Feb and enter back.

The risk you will have on the way to the immigration office is same to that another OP.

sample photo of the stamp by Tod is in this comment.

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Offhand I'd say you should be ok.