The Ark could have been repurposed at some point or not. I honestly have no clue. I just know Noah and his family did not do it according to the bibles narration. First we have to identify where the Ark may have landed.
Scripture says thus Gen 8:4 And the Ark rested in the seventh month, on the seventeenth day of the month, upon the mountains of Ararat.
Ararat was a country also known as Urartu. The country did not exist when Moses wrote Genesis. Scholars find that name came later in the 6th or 5th century BC. So basically Moses would have written the name of that geographical area in his time in Hebrew. Later on when scribes were editing the writings of Genesis they would have not changed anything as the geographical area would be the same. However, since that country or confederation of countries no longer existed they would write the modern country at that time or edit the geographical name to be current.
What that then does is at that time Ararat was composed of several mountain ranges that would fit into the geography of the mountains of Ararat. Those ranges today are what we would identify as the Lesser Caucasus, Eastern Pontus, Armenian Highlands, Taurus Mountains, and Zagros Mountains.
We also know that when they came off the Ark that they were living in tents. Gen 9:20 And Noah began
to be an husbandman, and he planted a vineyard:
21And he drank of the wine, and was drunken; and he was uncovered within his tent.
My thesis would be they came off the Ark on one of the mountains in the country of Ararat, which several mountain ranges comprise the country of Ararat. They were nomadic living in tents and were southward bound. They are going to have a great vantage being in the mountains and coming down the ranges to view all the land to the west of them. At some point down in the southern Zagros Mountains they viewed a fertile plain and came down and made that home.
Gen 11:2 And it came to pass, as they journeyed from the east, that they found a plain in the land of Shinar; and they dwelt there.
They did not have any wood with them or use wood as Gen 11:3 says. And they said one to another, Go to, let us make brick, and burn them throughly. And they had brick for stone, and slime had they for morter.
So I don't find Noah or his family used any wood from the Ark for habitations. As scripture just paints a different picture. Did someone later I have no clue. So could it have been, there is a possibility at a later date. It just wasn't upon the exit and the first habitation they built upon leaving the Ark.