Kitchen sink backing up problem
There's a blockage farther down the line somewhere. It is most likely right where the trap arm connects to the vent and vertical drain. Try to auger this fitting out. Access by removing the trap and hopefully the adapter that turns down into the trap. If the adapter does not come off, you'll have to work through it's bend.

If you can't get a clean running drain by working at this fitting, it is time for professional assistance.