Hello, sorry for my english.

I used:
Joomla! 3.2.4 Stable
K2 v2.6.8
JCE 2.4.2

Problem:
I've created a few additional fields (text area with the use of the editor). While filling in the large amount of information, it is not saved, as well as all the additional fields are set to zero. If the information in the field is small, then everything is fine. Changed editor, did not help.

Maybe somewhere there is a limit to the number of characters?

This sounds a DB limit issue.
IT probably fills up the database cell and it reverts to the default values instead of being saved.

How 'large' is the amount of information?
If you want to use a really large text (larger than the intro/full text) then you should try to change the xx_k2_items -> extra_fields cell to medium/ large text instead of text, like the other text fields.

I am trying to enter text in the amount of 20880 characters. If I enter in less than half, then the field is successfully preserved.

It is a bit too much for an extrafield.
You can either use the fulltext for this or try my suggestion in my previous post, just be sure to backup your BD before doing anything like this.

Thanks for the help, I'll try to change the xx_k2_items -> extra_fields cell.

Let me know how it went.
Also backup first :)

I found xx_k2_items -> extra_fields cell and replaced type of text on mediumtext (tried and longtext), but still the problem persists :(. Unfortunately I can't use introtext and fulltext, as they have been used already.

Make sure you delete any browser and Joomla! cache before you try again.
Log out and in again to clear any sessions as well.

If it still does not work then you might need to implement a K2 plugin to split the large amount of text handled, or swap the fulltext with an extrafield.

Is there a link I can see to the site?