Does anyone in Tanach ever get divorced? Not merely separated but fully divorced.
Didn't Moshe Rabbeinu get divorced from Zipporah? See Rashi Bamidar 12:1,
על אודות האשה: על אדות גירושיה.Sounds like he divorced her. I don't think this shittah is universal though. Still looking for more sources.
Tosafos in Yevomos 62:a
dichsivsays that possibly he wrote her a Get.
Pirkei d’Rabbi Eliezer (chapter 30) brings down the following story that records that Yishmael divorced his wife.
Once Avraham wanted to visit Yishmael to see how he was doing. Sarah okayed his visit on condition that he didn’t get off of the camel. So Avraham traveled through the desert to the home of Yishmael. When he arrived, Yishmael’s wife, Eysa, told Avraham that Yishmael had gone with his mother, Hagar, to gather dates in the desert. Avraham said he was weary from his journey, might he be served some bread and water? Eysa refused, claiming she had none.
“Please tell your husband” Avraham told her, “that an old man from Canaan came to see him. Tell him his threshold is broken.”
When Yishmael returned, his wife gave him the message. He understood that his father had attempted to visit him, and he understood from his message how cruel his wife had been. Yishmael divorced his wife.
(Sotah 12:a) is Amram who divorced Yocheved before remarrying her upon Miriam's advice. In fact, the Gemoro relates that the whole nation followed suit and divorced their wives. (Albeit, they [including Amram] all remarried soon after!)
King David's wife Queen Batsheva was previously married to Uriah. Uriah wrote a Get to her before he went to war so she should not be an Agunah if he were not to return. This Get came into effect at Uriah's demise, itself brought about by the King. (Kesubos 9:b)
The only divorce I can find in Tanach al pi peshat is Avraham's divorcing Hagar. The verse (Genesis 21:10) says:
גָּרֵשׁ הָאָמָה הַזֹּאת, וְאֶת-בְּנָהּ
Cast out this bondwoman and her son.
The word used is גרש which is the word used for divorce generally in Tanach (eg. Leviticus 22:13) and it seems to be the peshat here because we never hear of Hagar again.
For what it's worth, pseudo-Jonathon explicitly interprets the verse: ופטרה בגיטא - and he exempted her with a document.
Hashem told Hoshea to get divorced
Adam was given a get like a woman (Bereishis 3:24 - Vayegaresh Et HaAdam, literally "and He (God) divorced the man") Opening of Tana D'bei eliyahu
This probably has kabalistic meanings