According to this blogpost, it depends on whether the game is postseason or not. For the postseason, the game would continue for a second period of overtime with Team B in possession, but for the regular season the game would end 23-20 for Team A.
Of course a blog isn't always the most reliable source, but we can check this also in the rulebook: checking the link you sent regarding Rule 16, there are different articles for preseason/regular searon and for postseason.
ARTICLE 4. OVERTIME IN PRESEASON AND REGULATION SEASON
The following shall apply to overtime games in the preseason and regular season.
There shall be a maximum of one 15-minute period, even if the second team has not had an opportunity to possess the ball or if its
initial possession has not ended. If the score is tied at the end of
the period, the game shall result in a tie.
The only sensible interpretation of this is that in regular season Team A would win for your scenario. Team B cannot get the ball because there can only be one 15-minute period, and the game cannot be a tie because the score is not tied.
And as Dr.DrfbagIII pointed out in a comment, this would be the place to point out explicitly any extraordinary rules why 23-20 would magically turn into 23-23, and in the absence of such a rule we can assume that Team A wins 23-20.
ARTICLE 5. OVERTIME IN POSTSEASON
The following shall apply to
overtime games in the postseason:
If the score is tied at the end of a 15-minute overtime period, or if
the second team’s initial possession has not ended, another overtime
period will begin, and play will continue, regardless of how many
15-minute periods are necessary.
Note the phrase "if the second team’s initial possession has not ended". This means that in postseason Team B would be given a chance to score too.
As to the question of whether this has happened, I am not sure but would guess not. According to the same blog, there has only been one drive over 13 minutes in modern NFL history (13:27 by Tennessee Oilers vs Dallas in 1997). Given the relative rarity of overtime drives, and that the new overtime rules have only been used in the last couple of years, I guess that it hasn't yet occurred.