Finger and whole hand is swollen and red from a wooden splinter?

Serious infection. If the whole hand is red and swollen from a splinter , most possibly you have a serious infection and need immediate attention. Please go to an er or see a doctor asap.
May be cellulitis. Besure to have it seem by a doctor. The splinter could have carried an infection into the finger and now it has spread to the whole hand. In the mean time, warm compressed and draining any puss at the splinter site. And remove the splinter if you havent done so yet.

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