This might not be the answer, but for reference here is a problem that can occur if the plumbing is not installed properly.
If you look at Fig. 3, this is what a proper drain looks like. You'll notice the orange line represents the water level in the system, the water levels out in the trap preventing sewer gases from entering the sink drain.
In Fig. 1, you can see what will happen if the drain line has to go up before meeting the main drain line. Again the orange line represents the water level in the system, and as you can see the water level is much higher in the drain.
Fig. 2 shows what could happen if the sink was installed lower than the rise in the drain line. The water would never fully drain from the sink, since it would require extra pressure in the system to push the water up the drain.
![enter image description here](https://i.stack.imgur.com/Ws9y4.png)
I wouldn't guess this is the issue in this case; since as you can see, if you opened the trap in this case you would have more water drain than what is typically in the trap.
In a perfect world Fig.1 and 2 would never pass inspection, but as we all know not all plumbing is properly installed and/or inspected. So this could possibly be the case in this situation (given the plumbing was installed by an amateur plumber), but it is doubtful. It would be more likely caused by a clog in the line.
I would start by removing the trap, putting a bucket underneath the tailpiece (below the sink) and just running some water and seeing what happens. Clean out the trap while you're at it and make sure there isn't a mess of hair stuck in there. If it all comes through without a problem and doesn't back up, then I would suggest buying a snake and going through the wall pipe and seeing what you can pull out. I've always been able to get a slow drain going with a snake. It might take a few tries, but it can usually work out. Make sure that you push the snake right up against the pipe opening, and then just keep cranking until something changes.
You want to be careful with chemicals because they can discolor the chrome around the sink if you're not very careful, and they can also react with the pipes in a unsafe manner. They make some gels that are more organic in nature that can eat away at organic clogs (hair, etc) that are much safer to use.
Best Answer
Assuming there are no other drain problems occurring (because you did not mention any) you most certainly have an obstruction of the branch drain line leading to that sink.
Option 1- run a good quality drain auger down the drain, through the trap, and down the branch line as far as it will go.
Option 2- take the trap back apart and run the cable directly down the drain line.
For either option, it is important that you use the correct technique and persevere until the length of the cable has gone down. It's not easy, be prepared to work. There are numerous instructional videos on-line, this one shows the correct technique but the person is using a cheap sub-standard auger.
I always cable from the top and run water until it backs up a bit, so I can tell when I have hit the clog (because the water suddenly drains). I highly recommend that you invest in a good professional quality auger, I did and it has saved me hundreds and hundreds of dollars over the years:![enter image description here](https://i.stack.imgur.com/gxBhJ.png)
Always wipe off the cable with a rag as you remove it from the drain, and always spray it down with your favorite water displacing spray lube that comes in a blue and yellow can or it will rust in the drum.