Can I apply for permanent residency in Thailand after three years on a retirement visa extension?

Okay, you can't apply for permanent residence off ANY visa (the kind you border bounce on to get another 90 day stamp) it has to be on yearly extensions of stays and they cannot be broken during the 3 years prior to you applying.

Also you can't ever qualify for permanent residence on what's called a "retirement visa" because you can't hold a work permit and legally work on one.

If you are married to a thai, legally working here you can after a period of years apply directly for thai citizenship and totally skip the permanent residence process.

This is a LONG but good topic about how to acquire permanent residence

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To apply for Thai Permanent Residency, you need to fulfill the following criteria:

You should have had a Thai non-immigrant visa for at least 3 years prior to the submission of your application.

Holders of multiple Non-Immigrant visas can not apply.

In order to qualify you must have 3 consecutive yearly extensions.At the time of the submission of application, you must be a holder of a non-immigrant visa.

You must be able to meet one of these categories to apply for PR status in Thailand:

Investment category (minimum 3 – 10 Mil. Baht investment in Thailand)

Business / Working category

Support a family or Humanity Reasons category: In this category, you must have a relationship with a Thai citizen or an alien who already possess a residence permit as a husband or wife; father or mother; or a guardian of a Thai child under 20 years of age.

Expert / academic category.

Other categories may be determined by Thai Immigration.

As Ron said, for retirement.....no.

In a word....no. You need to of been working and been paying taxes for a solid 3 years before even beginning the process.