Are there any examples in the Tanach where somebody is being cynical?
Cynical: Definition; from The Merriam-Webster dictionary:
Having or showing the attitude or temper of a cynic: such as a : contemptuously distrustful of human nature and motives … those cynical men who say that democracy cannot be honest and efficient. — Franklin D. Roosevelt b : based on or reflecting a belief that human conduct is motivated primarily by self-interest - a cynical ploy to win votes
Example in Tanach:
Shmos 1:9-11, Shmos 5:8, Shmos 7:10-13
Pharoah is being pretty cynical towards the Jews.
1: (9) "He said to his people, "Behold, the people of the children of Israel are more numerous and stronger than we are." (10) Get ready, let us deal shrewdly with them, lest they increase, and a war befall us, and they join our enemies and wage war against us and depart from the land." (11) So they appointed over them tax collectors to afflict them with their burdens, and they built store cities for Pharaoh, namely Pithom and Raamses."
5:(8) "But the number of bricks they have been making yesterday and the day before yesterday you shall impose upon them; you shall not reduce it, for they are lazy. Therefore they cry out, saying, 'Let us go and sacrifice to our G-d'."
7:(10) "Moses and Aaron came to Pharaoh, and they did so, as the Lord had commanded; Aaron cast his staff before Pharaoh and before his servants, and it became a serpent. (11) Pharaoh too summoned the wise men and the magicians, and the necromancers of Egypt also did likewise with their magic. (12) Each one of them cast down his staff, and they became serpents; but Aaron's staff swallowed their staffs. (13) But Pharaoh's heart remained steadfast, and he did not hearken to them, as the L-rd had spoken."
King David admits to being cynical when he was suffering:
(11) "I said in my haste, "All men are liars."
Rashi: I said in my haste: to flee from Avshalom (his son).
“All men are liars”: Heb. כזב, [meaning that one] betrays his friend, for I saw my son betray me and seek my life, and [I saw] all Israel repay me with evil for good; therefore, I believed Ziba and said that Mephibosheth also lies and betrays me. Every expression of כָּזָב is purely an expression of lack of trust
The Satan is being cynical:
(8)Now the L-rd said to the Satan (adversary), "Have you paid attention to My servant Job? For there is none like him on Earth, a sincere and upright man, G-d-fearing and shunning evil." (9) And the Satan answered the L-rd and said, "Does Job fear G-d for nothing? (10) Haven't You made a hedge around him, his household, and all that he has on all sides? You have blessed the work of his hands, and his livestock has spread out in the land. (11) But now, stretch forth Your hand and touch all that he has, will he not blaspheme You to Your face?"
The Baal HaTurim points out that there are only two times in all of Tanach that “vaYivez” [and he mocked] is written. “vaYivez Eisav the Bechorah” [Shmos 24:34] and “vaYivez Haman to merely send his hand against Mordechai alone” [Esther 3:6]. The Medrash calls Haman a “mocker the son of a mocker” (bozeh ben bozeh). He was a denigrator the son of a denigrator, a “letz” the son of a “letz”, a cynic the son of a cynic.
Also, Rav Zivotofsky says,
One of the most popular phrases is “Eretz zavat chalav udevash,” which appears both in Chumash and Nach.2 This phrase is first used to highlight the fertility of Israel as compared to Egypt (Exodus 3:8 and 17). Subsequently, Korach refers to it cynically to describe Egypt (Numbers 16:13).
This case combines cynicism with sarcasm:
Kings I 2:22 Adonijah, King Solomon's brother, would like to have Abishag the Shunamite for himself. He requests Solomon's mother, Bat Sheva to relay this message to King Solomon. The King's response is, "Why does he ask for just Abishag? He should have the entire kingdom, too, for, after all, he is my brother!"
This case sounds cynical:
Kings II 5:7 Na'aman, the king of Aram, was leprous. He sends a note to the King of Israel saying that he will be sending a messenger to him asking for a recommendation on how to cure him from his leprosy. When the king reads the letter, he tears his clothes and says, essentially, that the only reason he is sending this letter is because he knows that I can't perform this, and they he will use this as a pretext to battle against me and my people.