I am adding outlet in my kitchen. The circuit is 20 amp and 12-2 wire.
Can I add 14-2 wire off 20 amp outlet to new 15 amp outlet?
It is only 3 feet between outlets.
Thanks in advance.

You will be limited to a 15 amp breaker. Just run #12 and it could be on a 20 Amp breaker.
You can still use your 15 amp receptacle. This will go to the load side of the Recepticle(GFI), if no GFI in Panel.

If you are adding this receptacle to one of the two mandated 20 amp circuits for the kitchen, the answer is no. You would only be allowed to extend the circuit with #12 AWG conductors.

It is only 3 feet between outlets

A receptacle outlet shall be installed so that no point along the wall line is more than 24 inches measured horizontally from a receptacle outlet in that space.

Just to add. Kitchen receptacle outlets shall be located above, but no more than 20 inches above the countertop.

What does this mean: Exactly what it says. It also means you can have a receptacle outlet 3 feet away a receptacle outlet in that space. Key:No point along the wall line

Thanks for all the answer!
My house is 40 years old and I see some wires labeled 14-2 NM type, but it is thicker than today's romex 14-2.
Anyone know why? Can old 14-2 wire handle more than 15-amp because it is about same thickness to today's 12-2..

NO! #14 is the gauge of the conductor. The rest is insulation. Older insulation is much thicker than today's.

Keep in mind, the requirements laid out by Washington are the minimum requirements for new installs or remodels of required kitchen counter receptacles.
If you are just adding a receptacle you can put it basically where ever you want.

But even wire conductor is thicker? Thanks.