yes, but the question was, can you do it in Chile for 6 months, and that you cant.
you can leave without your bike, no problem, but you must come back and leave the country with your bike before the bikes 90 days are over.

Yes you can get an extension. You just fill out the form and turn it into Aduana and get the approval. I have been told you can do this on the internet as well.

My recent experience in Chile with a motorbike overstaying past the 90 days granted: I entered Chile in mid-Oct 2017 on my US registered moto through Los Andes; I was granted permission to keep the bike in the country until mid-Jan 2018 (same as my tourist visa). I flew back to the US from Concepcion for 2 weeks around Christmas time (leaving the moto behind at a friend's house - no questions were asked about my vehicle when I flew out - nor did I mention it to anyone on my way out ).

When I returned I was given a new tourist visa for 90 days ... I completely forgot about the moto needing to leave the country by mid-January

In mid February I reach the Chile Chico border. I got my passport stamped but then Aduana noticed the paperwork for the bike expired a month prior. It caused a lot of confusion and with my broken Spanish I tried to explain that I flew home for Christmas and was granted an additional 90 days - played dumb and said I thought the bike was just linked to my visa's timeline. The Aduana line started piling up behind me (which might have worked to my benefit!). After a bit of back and forth between Aduana and PDI in the end they just gave me a stern warning that I shouldn't do it again and stamped my papers and set me free. No extra paperwork required!

As a side note: they told me there were no problems to re-enter the country a few days later (as I was heading south into Puerto Natales ... which then gave me a new 90 days on the moto).

Out of curiosity does anybody know what I narrowly avoided? What are they supposed to or able to do when your vehicle overstays? Fine you, confiscate the vehicle?

I also overstayed the TIP of my bike (norwegian plated) June/July 2017 in Chile. Due to mechanical problems and endless waiting for spare parts which for the most part never arrived I overstayed the TIP. The mechanic that helped me with my bike phoned the Aduana and explained the matter and was told to write an email where we explained the situation. So an email was written and sent - and after several weeks more of waiting I was given the fine of 20 000 chilean pesos which is around 25 € I belive and the permission to leave. I was told I was lucky as the fine should have been 10 times more. The initial overstay was only 1-2 weeks.

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In the end everything will be fine. If its not fine its not the end....

I rode to Ushuaia from the US in 2015/16. I met the love of my life in Santiago so now I reside there 6 months of the year and spend the other 6 months in the US. When in Chile I make sure to run across the border before my 90 day motorcycle visa expires. I just spend one night in Argentina then cross back. For the past 2 summers while in the US, I was able to get extensions to my visa until I returned. You can do that online. The first time I did it I found a nice person in Aduana who was very helpful in telling me the procedure, what form to fill out etc. The second time that person had moved on to another department and the new person was not familiar with the procedure and it was more challenging. It helped greatly having a local with me to translate. You need to email the completed request form along with copies of the entry documents, passport, and motorcycle documents. Do it at least two months before the expiration date as it can take time. Apparently though, you are really supposed to visit the Aduana at the airport before flying out of the country and fill out all kinds of paperwork there. I never did. If your motorcycle exceeds the 90 day limit they can confiscate it. Now I am married and working on a temporary residency visa. Once I get that I will try to get a visa for the moto to cover it for the same time frame. I've been told it is not possible to import a used moto or car but there are loopholes which I hope to discover. If not, I will sell my baby to a gringo and buy a local bike. They are more expensive here but can be found.