I've got a splinter in my finger today in camp, and the medic told me that it should come out on its own after two-three days to one week. Some people thought I should take it out with tweezers after making a cut with a needle, but others insisted it will come out on its own. It's a bit deep in.
If it is superficial, it is not really concerning as yes, it will over time come out on its own. If it is deep as you said, the main concern would be for infection as wood is a porous material and it would not be ideal to be left in a deep wound. Did you bleed at all when you got the splinter? If you did, I would consider it deep enough to be problematic and get a medical professional to remove it.