Can I extend my Chile tourist visa?

If you need to request an extension, you may do so at any Departamento de Extranjería y Migración.

You need to present:

• Photocopies of the valid passport (identification pages, number and, granting and expiration date, plus the page with the seal or stamp of the last entry to Chile). Do not forget to present the original passport.
• Photocopy of the valid Tourism Card.
• A fee of US$100 equivalent in local currency regardless of their nationality

An alternative to that is leaving Chile briefly and re-entering (known as a visa run). Based on the link in Nathan's answer, it seems to be a popular method but there's no mention of how kindly Chilean authorities will look to that.

I performed the visa extension recently (March 2019) in Santiago and here are some Santiago-specific details to add to the accepted answer which is still largely correct.

• The official guide mentions what you need: photocopy of: passport photo page, passport page with Chilean entry stamp, copy of "Tarjeta de Turismo" which a receipt with the words "PDI" on it given to you on entry (you should always keep this in your passport. I had these photocopies, but it seems like the agent would have been willing to copy them for me if I didn't have them (but don't rely on it).
• In principle you are supposed to book an appointment at this website, but in practice everything is booked up due to the huge influx of immigrants from other South American countries, so you may be waiting weeks or months on the waiting list of an appointment. Since you can only apply for the extension in the last 30 day of your 90 day initial visa, making an appointment online is pretty much impossible unless you get lucky with the timing.
• Instead, simply go to the immigration center at Matucana 1223, some time before 11 AM. There will be a crowd of people at the door, mostly entering with their appointments booked online. Make your way to the front of the crowd and tell the person taking appointments that you want "prorroga de turismo" and that you are leaving Santiago soon and won't be able to return to an immigration office soon.
• You will be let in and have to wait for an appointment (in my case about 15 minutes) and then the process is more or less as the accepted answer describes. Overall it took me about 2 hours after being let in the first time.
• The main problem is if they don't let you in the front door. They may ask you how much longer you have on your current visa and tell you to come back another day. If you make it clear you cannot, they should let you in. You can also sneak in the exit door if it is unguarded, which is at the back of the building directly opposite the entrance (walk around the block to get to it), once inside ask anyone official about the "prorroga de tourista" and they should direct you to the right place.

If you decide to stay in Chile for more than 90 days as a tourist, you have three options: obtain a visa, buy a 90 day extension for $100, or leave Chile and come back. For many people the latter is the best choice, as the narrow shape of Chile makes it a short trip to leave the country.

Obtaining a visa extension:

You may receive a one year tourist visa with the proper paperwork. It is not difficult to receive, but you must prove that you will not be a financial burden to the country. Copies of bank records showing sufficient funds are helpful, but they may ask you what activities you are doing in Chile. It may be necessary to provide a certificate proving you are doing beneficial work in Chile, even if it is volunteer work. This is not set in stone, and different people have had different results. If you are doing nothing in Chile that is deemed beneficial, and are asked for a certificate, you can volunteer to teach English in any school, even if it is only for a few weeks.