Oh my God! This is so creepy because the same thing just happen to me like ten minutes ago! So, so weird!
Well, I happen to be having some thin paintbrushes near by and I just used those. If you have anything similar, that will also work. Any two thin objects. Try to take it out s-l-o-w-l-y or else it will just get stuck further.
P.S. I hope that helped. I still think it’s an awkward coincidence.
Go under the sink and undo the trap in the drain pipe. These things are normally just hand tightened and should come apart fairly easily, unless yours has been there forever. Loosen the pipes and take the trap off, put a bucket under the open drain, and flush some water into the sink. Then just reassemble the trap.
You probably will only push the cap farther down if you stick anything down the drain to try to retrieve it. The cap is probably wet and will not stick to chewing gum.
PS: when you take the trap apart, have a shallow pan of some sort underneath. The trap is full of water (and gunk).
There is really a simple solution. Look under the sink. Do you see the drain tube from the bottom of the sink going down into a U shape and then into the wall ? There should be a flange at each end of the U shaped trap that connects to the sink tube and the drain pipe with a threaded nut. Loosen the nuts at each end of the U trap and it will slide off of the sink drain and the main drain. BE CAREFUL !! If this drain assembly is really old it will be brittle and could break. So don’t get rough with it and the trap will have water in it so keep a rag or bucket handy. Once you get it off you can just dump it out and if the cap is in the down tube from the bottom of the sink just use something to push it on out the bottom. Reassembly of the trap and drain is a reverse of the removal proccess.