It remained afloat and drifted to USS Tennessee during the attack and remained with Tennessee until Tennessee was freed from her berth a few days later. Judging by Matt's photo, it doesn't appear that the initial Arizona explosion is the cause of the damage to the Lighter.
Nevada's deck logs state that several bombs missed her Portside but no mention of any to Starboard. She took one bomb on her bridge before she sortied.

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if still afloat then not hit by a bomb as would have blown the lighter apart & sink it. even a dud would cause a hole in the hull & sink it. if still afloat after attack then possibly hit by wreckage from downed aircraft or from the Arizona when she blewup.

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Going back to the discussion about the damaged lighter on USS Tennessee's Starboard beam. We discussed it on May, 17th, and now can be found on page 6 of this forum.
Taking another look at this photo that Matt posted clearly shows that USS Oklahoma has already capsized and USS Arizona has already exploded and the lighter on USS Nevada's starboard beam is very much undamaged.
I recently came across this shot showing Nevada's sortie. Here you can see she is just passing USS West Virginia. USS Arizona is in the middle of the shot, and the Lighter that USS Nevada left behind is still positioned in-between the two F-8 mooring quays. Only now, you can see the roof on the Lighter is very much damaged and blown outward.
This begs the question if a USS Nevada near miss bomb did indeed strike this Lighter.