Southern Aravah

Answer

Olei Bavel territory includes most of Israel, and shemitah laws apply in full. Olei Mitzrayim includes land in southern Israel, where shemitah laws are less strict. However, the Mishnah (Shevi'it 6:1) mentions yet another area in the Land of Israel where shemitah laws do not apply at all. This is gevulot hahavtachah, the land promised to Avraham Avinu from "the River of Egypt until the Euphrates." 

Are there areas in the State of Israel that are outside of olei Mitzrayim but part of gevulot hahavtachah, where shemitah laws do not apply at all? 

There is a major dispute about this. The Chief Rabbinate of Israel and the rabbis of Torah VeHa'aretz Institute maintain that the entire State of Israel is part of olei Mitzrayim territory. However, there are lenient opinions that hold that parts of the Aravah are outside of olei Mitzrayim. There are many Badatz certifications that rely on this leniency. That means that even though they don't rely on heter mechirah, they allow Jewish farmers in southern Israel to plant vegetables during the shemitah year and sell this produce as mehadrin.
(In practice: farmers in these areas also sell their fields to non-Jews, because of the requirement of the Chief Rabbinate, not because the Badatz requires it).
Often, this produce from Southern Aravah is sold under the label יבול חו"ל , imported produce, to avoid confusing consumers. Nevertheless, by buying Aravah produce, we also support Jewish farmers. 😁 

Note that there is not extensive farmland in this area, so the amount of produce is limited.

There are four major opinions on the line that marks the border where south of it shemitah laws do not apply (see map: https://en.toraland.org.il/media/5271353/map-olei-mitzraim-and-bavel.pdf):

(1) Some hold that it is south of Wadi Tzin (Tevout Ha'aretz, Rabbi Joseph Schwarz) - northern Arava.

(2)  Others believe it is further south (Badatz Eida Charedit, Admat Kodesh Rabbi Yitzhak Goldhar) - Middle Aravah.

(3)  Some hold that Southern Aravah is below 30°N, some 50 km north of Eilat, based on Rambam in Hilchot Kiddush HaChodesh (Rabbi Efrati in the name of Rabbi Elyashiv and the Chazon Ish) - southern Aravah.

(4)  Rabbi Tucazinsky and the Chief Rabbinate hold, as mentioned above, that olei Mitzrayim goes all the way down to Eilat. This is the stance of the rabbis of Torah VeHa'aretz Institute.

Besides this, some hold that east of Nahal Aravah is considered gevulot hahavtachah (Badatz Eidah Charedit).

See here a map.

For a more detailed discussion of the major opinions and demarcations of the various territories, see The Consumer's Guide to Shemitah, Chapter 11, here:  https://en.toraland.org.il/beit-midrash/articles/shemitah/buying-fruit-and-vegetables-otzar-haaretz/supplying-agricultural-produce/