For Christ also suffered once for sins, the righteous for the unrighteous, that he might bring us to God, being put to death in the flesh but made alive in the spirit, in which he went and proclaimed to the spirits in prison, because they formerly did not obey, when God’s patience waited in the days of Noah (Genesis 6-8), while the ark was being prepared, in which a few, that is, eight persons, were brought safely through water. Baptism, which corresponds to this, now saves you, not as a removal of dirt from the body but as an appeal to God for a good conscience, through the resurrection of Jesus Christ -- 1 Peter 3:18-21 (ESV)

Some argue Peter meant that "Baptism" "correspond" or "represent" to Noah's family's experience which could implies that Noah's family was not really "baptised" especially if you consider how modern Christian churches do baptism these days. But if you include the former verses the subject is focussed on the "death in the flesh but made alive in the spirit" which "Baptism" more likely represents. This could mean that Peter might have implied that Noah's family were also "baptised". Does that mean Peter claimed that every person that goes underwater in a submarine is now considered formally baptised?

Paul makes a direct reference to Baptism during Moses lifetime:

For I do not want you to be unaware, brothers, that our fathers were all under the cloud, and all passed through the sea, and all were baptized into [among] Moses in the cloud and in the sea, and all ate the same spiritual food, and all drank the same spiritual drink. -- 1 Corinthians 10:1-4 (ESV)

(Note, that the Greek word translated as "into" could also mean "among" or "same place/time" of Moses. Since Jesus told his disciples to baptize people in his name and not in Moses' name, Paul more likely meant "among".)

Baptism was already a familiar concept in Ezekiel's lifetime, although it wasn't called "baptism":

Then I will sprinkle clean water on you, and you shall be clean; I will cleanse you from all your filthiness and from all your idols. I will give you a new heart and put a new spirit within you; I will take the heart of stone out of your flesh and give you a heart of flesh. I will put My Spirit within you and cause you to walk in My statutes, and you will keep My judgments and do them. Then you shall dwell in the land that I gave to your fathers; you shall be My people, and I will be your God. I will deliver you from all your uncleannesses. -- Ezekiel 36:25-29 (NKJV)

However, some scholars argue this is a prophecy about a future "baptism" that would not have existed yet in Ezekiel's time.

Then there is also the story of Naaman:

Now Naaman, commander of the army of the king of Syria, was a great and honorable man in the eyes of his master, because by him the LORD had given victory to Syria. He was also a mighty man of valor, but a leper.

And the Syrians had gone out on raids, and had brought back captive a young girl from the land of Israel. She waited on Naaman’s wife. Then she said to her mistress, “If only my master were with the prophet who is in Samaria! For he would heal him of his leprosy.” And Naaman went in and told his master, saying, “Thus and thus said the girl who is from the land of Israel.”

Then the king of Syria said, “Go now, and I will send a letter to the king of Israel.”

So he departed and took with him ten talents of silver, six thousand shekels of gold, and ten changes of clothing. Then he brought the letter to the king of Israel, which said, Now be advised, when this letter comes to you, that I have sent Naaman my servant to you, that you may heal him of his leprosy.

And it happened, when the king of Israel read the letter, that he tore his clothes and said, “Am I God, to kill and make alive, that this man sends a man to me to heal him of his leprosy? Therefore please consider, and see how he seeks a quarrel with me.”

So it was, when Elisha the man of God heard that the king of Israel had torn his clothes, that he sent to the king, saying, “Why have you torn your clothes? Please let him come to me, and he shall know that there is a prophet in Israel.”

Then Naaman went with his horses and chariot, and he stood at the door of Elisha’s house.

And Elisha sent a messenger to him, saying, “Go and wash in the Jordan seven times, and your flesh shall be restored to you, and you shall be clean.” But Naaman became furious, and went away and said, “Indeed, I said to myself, ‘He will surely come out to me, and stand and call on the name of the LORD his God, and wave his hand over the place, and heal the leprosy.’ Are not the Abanah and the Pharpar, the rivers of Damascus, better than all the waters of Israel? Could I not wash in them and be clean?”

So he turned and went away in a rage.

And his servants came near and spoke to him, and said, “My father, if the prophet had told you to do something great, would you not have done it? How much more then, when he says to you, ‘Wash, and be clean’?”

So he went down and dipped seven times in the Jordan, according to the saying of the man of God; and his flesh was restored like the flesh of a little child, and he was clean.

And he returned to the man of God, he and all his aides, and came and stood before him; and he said, “Indeed, now I know that there is no God in all the earth, except in Israel”

-- 2 Kings 5:1-15 (NKJV)

What is immediately striking is that:

• Naaman had faith when believed that "the Abanah and the Pharpar, the rivers of Damascus, were better than all the waters of Israel", so going to the Jordan was very inconvenient to him and required an act of faith.
• Naaman "dipped seven times in the Jordan"
• Naaman converted when said: "I know that there is no God in all the earth, except in Israel"
• Although not explicitly stated, one can assume that he repented from sin when he realized that Israel's God is the true God.

However,

• Nobody publicly declared the Trinitarian baptism formula.
• John the Baptist and Jesus was not born yet, so neither of them could have instituted "baptism".
• There were no other priest or even the prophet himself around to perform "the baptism".
• The original intention of Naaman was to receive physical healing and not spiritual healing, however as a side effect one could argue that he did receive spiritual healing too.
• Some would argue the order is incorrect and that one could only be baptized after repentance and not the other way around.
• Naaman's actions might have nothing to do with baptism at all. It could mean that Naaman was obeying the law of Leviticus 14:1-9, but then one could also argue that some elements of that law is also lacking in Naaman's actions.

Although not directly connected to "baptism", the concept of physically washing to get rid of sin already existed:

Wash yourselves; make yourselves clean; remove the evil of your deeds from before my eyes; cease to do evil, learn to do good; -- Isaiah 1:16-17 (ESV)