You may need to get someone to try and get the piece of wood out of your foot. Oops, it has just occurred to me that it may be a plastic toothpick and not a wooden one!
Use a sterilised thin needle or pin to help get it out.
I presume that a piece of the toothpick has broken and some of it is stuck in your foot. It needs to get removed and if there is no one at home to help you to get it out (depending on which part of your foot it has gotten into), you may need to make an appointment to see the nurse so that she can get that piece out for you.
Sometimes the body has a way of getting rid of small things like splints and an infection with pus may occur and then the piece will be pushed out with the rubbish. Other times the skin can grow over the piece and it will be in your body. Many years ago I got a very large and thick cactus thorn go into the top of my middle finger under the first knuckle. The doctor said that it was too deep to get it out and it was best to be left where it was. That was 30 years ago and it is still in my finger, I have a small lump in that place and it still hurts when pressed.
After you soak you foot and still have difficulty in getting the pick out, put a small ring bandage (or a corn plaster with a hole in the middle) around the pick so that it does not get more deeply imbedded and make an appointment to see the nurse to get it out for you.
Magnesium sulphate paste (which is Epsom salts I believe) is a good drawing ointment, but this past cannot be used on broken skin.
Your event reminded me of my childhood-growing up in Crete by the sea,
where millions of sea-urchins were living. You know those spiky dark purplish to black colored round sea creatures?
Many a times we would accidentally step on them, ouch!
Necessity is the mother of invention: we would get a needle and then using the flame of a bic lighter we would burn the tip, apply some olive oil on the affected foot and after putting some alcohol on a cloth or a cotton ball, we would wipe the needle clean. We were then ready to perform surgery, lol!, by applying pressure around the surrounding area of the broken spike(s) in the foot and poking at it very carefully with the needle, until it would come out partially.
Small tweezers were very handy, if and when available, to catch the pertruding part of the broken spike and pull it out completely.
I must have removed hundreds from my own feet and my friends' as well.
Olive oil helped a lot to draw it out.
It was critical that this was done soon enough, since the spike was live and it would start travelling up the blood vessels and get in deeper into the body, as we were told. Yikes!
At least in your case, as Jemma said, the body has a way to get rid of it,
being a piece of wood.
Ouch! Did you not think of wearing any footware to avoid getting "spiked"?
Not sure if those plastic beach shoes were available when we were young :)
After I got my cactus thorn in my finger, I decided not to grow any cacti after that (this particular one was quite big and I was proud because I had grown it from seed); these days I get rose thorns stick into me (especially when I am idle and don't put on my thick leather gloves for protection) and then it is out with the needle to poke it out :)
We were just kids at the time, fooling around.
Times were tough then, we were happy to have a pair of shoes to go to school, never mind shoes for the beach,lol!
And people were tough then, didn't complain much and were easy to please.
I was more than happy with a slice of bread with olive oil drizzled over it & a pinch of sea salt. That was a real treat then!
We've come a long way.
I have a piece of a toothpick in the palm of my hand for over sixty years. No pain, but you can see it, or what is left of it.