First, Type about:preferences#advanced<enter> in the address bar. Under Advanced, Select Update.

[v56+] Type about:preferences#general<enter> in the address bar. Select Update.

Select Never Check For Updates.

Also turn off Use a background service to install updates

It doesn't work for me. Firefox updates itself despite selected options.

Do the options remain as you put them or change back?

https://support.mozilla.org/en-US/kb/how-to-fix-preferences-wont-save

Note: Some software, like Advanced SystemCare with Surfing Protection, can protect files in the Firefox profile folder against changes. If you have such software then check the settings or uninstall this software.

The options remain as I put them. "Reset" procedure doesn't help too.

I have no any software which protects Firefox profile against changes.

I called for more help.

FredMcD said

I called for more help.

Noone helped you?

I called again.

Maybe you have the Mozilla Maintenance Service installed on your system.

If that's the case :

To uninstall it, you'll need to scroll down to the section 'Uninstalling the Mozilla Maintenance Service' in this article  :

https://support.mozilla.org/en-US/kb/what-mozilla-maintenance-service

Thank you for your efforts, friends! It was helpful! Looks like it fixed the issue!

Here are all my steps that helped to fix the issue (I don't know clearly whether it affects or not but maybe someone find it helpful): 1. Uninstall Firefox (x86) 2. Restart PC 3. Uninstall Firefox (x64) yes, they were both installed on my PC 4. Restart PC 5. Uninstall Mozilla Maintenance Service (while uninstalling an error occured telling me that the service was already removed) 6. Install Firefox 55.0.2 and unselect checkbox to install "Mozilla Maintenance Service"! 7. During first Firefox startup go to Options imediately and verify that checkbox "Never Check For Updates" and "Automatically update search plugins" are unselected. 8. As you unselect Background update service during Firefox installation earlier there is no option "Use a background service to install updates". It's OK.

After all these steps performed it looks like it really doesn't updates itself. But I need more time to prove it.

Round of applause for you, Alexei  !

Keep us informed, will you please  ?

Next day it autoupdates again. I tried to monitor it by Procmon utility and finally I've figure out why it occurs. Accidentally I have found corporate protection system (Heimdal Security) which monitors all software installed on my PC and autoupdates it if necessary without any user confirmation. And there is no way to disable it because of group policy settings. So I found that its not Firefox issue but my environment issue.

Well done. Contact its support.