Outside of TV Tropes, the concept of Sudden Death is usually defined as "play that ends as soon as one competitor is ahead of the others". However, the article is pretty vague (only stating that this aspect "usually" applies) and suggests that the concept is wider than that without providing any usable boundaries.

Wick check results:

• 16 fit the strict definition
• 11 were "win this extra round and you win" regardless of whether it's possible to fall behind and later overtake the opponent during that round
• 6 emphasized a game putting extra pressure on players if they take too long
• 5 were other misuse
• 11 were ZCE or otherwise unclear

Possible solution: Rename it and expand it to cover tiebreaking rounds in general. This sidesteps the problem of defining what exactly counts as pulling ahead, and the Sudden Death flavour of tiebreaker doesn't seem common enough to warrant the hassle of keeping it as a sub-trope. "A game putting extra pressure on players if they take too long" could be spun off into its own trope, since not all of those apply to extra rounds.

• PyrrhicVictory.Live Action TV: In That Mitchell and Webb Look one episode of Numberwang went on for so long that the game went into Sudden Death. The first contestant to inhale enough of the poisonous Number Gas won. But, y'know, died. "The first to inhale poison gas" can be loosely considered an example of "the first to score a point wins".
• Funny.That Mitchell And Webb Look: Going for three whole days without hitting Numberwang. The Sudden Death round is literal; whoever dies first (while being bombarded with "Number gas") wins. It's made even funnier when Julie dies first, complete with "ding" and funeral flowers that spell out "That's Numberwang!" Reference to the above.
• Film.Sudden Death: Not only the title, but the hockey game going into this is plot-important. Not a very well-explained example, but the use of SD in ice hockey does fit.
• Own Goal: The 2001 UEFA Cup Final match between Liverpool and Deportivo Alavés is a rare occurence where an own goal is the deciding factor that determines the winner. Liverpool got a free kick in extra time, and Gary McAllister, their best dead-ball kicker at the time went on to make a shot. Alavés' player Delfí Geli attempted to block the shot with a header, but he accidentally put the ball into his own net. Liverpool ended up winning the match after that, thanks to the golden goal rule. Not explained, but the "golden goal" rule is an example of "first to score wins."
• Series.Where In The World Is Carmen Sandiego: If there is a tie in 2nd place after the final clue of round 1, They would go for it. The correct answer will advance to 2nd Round.
• Series.Family Feud: Present since the return to Single-Single-Double-Triple. If neither team has reached 300 after four rounds, a Sudden Death round is played with Triple point values and a question for which the #1 answer is typically in the 70s or higher. Whoever is first to ring in with the right answer wins the game. Could be better written, but it counts.
• EarlyInstallmentWeirdness.Game Shows: For Karn's first season, the Louie Anderson-era scoring system (single-single-single-triple, with only one strike in the Triple round) was retained. Karn was also less of a Large Ham on his earlier episodes, with much less shouting and reliance on his catch phrases. The "party" theme from the Anderson era was initially retained, even being given a remix, until the Combs-era theme and music package was brought back at the halfway point of the season. For Karn's second season, the scoring reverted to single-single-double-triple with a Sudden Death if neither team has hit 300 points, while the theme reverted to the previous "party" theme remix, which stayed put for the rest of Karn's run. Reference to the above.
• Series.Catchphrase: In the UK version, if the players failed to answer the Bonus Catchphrase when it was fully visible, they were shown one more and whoever got it won the round. If they didn't get that either, they continued playing sudden death until someone got one. (This did not apply to the US version; the money in the bank simply carried over to the next round.)
• UsefulNotes.Baseball: Sometimes happens in real life games — in baseball, extra innings go on as long as necessary, and are often referred to as "bonus baseball" or "free baseball". If the home team takes a lead in its half of any inning after the 8th, the game immediately ends, with no further play.
• Double Knockout: In the Super Smash Bros. series, it's not uncommon for multiple players to hit each other at the same time, which may potentially result in two or more players KO-ing each other. Due to the significant variability in launching speeds and blastline positioning however, it is very rare for two players who hit each other at the same time to be actually KO'd at the exact same time. Thus in a situation where both players are on their last stock and deal a KO-ing blow to each other, one would usually outlive the other by a split second and be declared the winner by the game. If two or more players are actually KO'd at the same time on their last stock, the game will proceed to Sudden Death. And if two or more players are again KO'd at the same exact time in Sudden Death, instead of going to another round of Sudden Death, the game declares the player with the lower port the winner in Melee, while deciding the winner randomly in the other Smash games. In a tournament setting the Sudden Death match is instead ignored and a draw game is decided by who was at the lowest percentage of damage/highest amount of stocks before the match ended. If both fighters were at an even percentage and stock total when the match concluded, then a quick one-stock bout with the same characters and stage is used as a more controlled version of Sudden Death. Long example that doesn't actually explain what it is. However, it's right, as the Smash series's idea of a sudden death round puts everyone at 1 stock and extreme damage that makes them get KO'd by a stiff breeze, which fits the spirit of the trope even if it's technically possible to pull ahead on damage.
• Series.Shop Til You Drop: In the event of a tie score at the end of the Shopper's Challenge round, one final question was asked. If the team that rang in answered the question correctly, they would advance to the bonus round. If they answered incorrectly, their opponents automatically won (fellow Stone Stanley game show Legends of the Hidden Temple also used this exact same tiebreaker format in its' first season).
• Series.Legendsofthe Hidden Temple: If two teams were tied for pendants after all three of Olmec's Temple Games, they go to the Sudden Death Podium and answer Olmec's final question to advance to Olmec's Temple Race. In the first season, if they were incorrect, their opponent automatically won. After that, the rule changed so that the other team would have to answer it correctly. The 2021 version keeps the same rules, but subtracts a Temple Game so teams can only win full pendants.
• Recap.Dragon Ball Tournament Arc: After Goku and Jackie Chun both take each other down with a Double Knockout, the tie breaker rules come into play; whoever is the first to stand up and declares themselves to be the winner, wins.
• Series.Taskmaster US: If two or more contestants are tied in first place, then a tiebreaker takes place. This only ever happened in one episode; "Die, Die, Die". It was a three-way tie between Rob, Kate, and Lisa. The task was to answer a question: For how many Sundays has the Taskmaster (Reggie, not the show itself) been alive for? Helpfully, a graphic on screen did the mathematics for each answer. Lisa wins, as her answer was closest to 2363 Sundays (45 years, 23 days)
• Memes.Game Shows: LONGSHOT! (screeeeeeeeeeech...CRASH!) ^Explanation What a Charger on Whew! would say when time was running out and s/he couldn't reach Level 6 in time. This forced a Sudden Death single-pick of three possible bloopers to decide the round.

• Laconic.Sudden Death: End-of-match Round allowing last-minute score to determine victory. Vague definition that doesn't say anything about the "first to pull ahead wins" aspect.
• VideoGame.Ultimate Chicken Horse: If two players were to win at the same time, those players have 60 seconds to finish the level again. First one to reach the goal wins. If you fall behind in the middle of that level, you can catch up again.
• Series.Have I Got News For You: In the early series, if the scores were tied at the end of the show there would be some sort of tiebreaker, which included using the caption competition (with the panellist who got the biggest laugh getting the point), the team captains playing Rock–Paper–Scissors, and an arbitrary general knowledge question being made up on the spot. The Angus scandal episode ended in a tie, but perhaps due to Deayton having other things on his mind no tiebreaker was played and it was the first ever episode to end in a draw. Only one episode since then has had a tiebreaker round: earlier in the show the Louis Tussaud wax museum in Great Yarmouth had featured as a news item as it was facing closure. Since the museum is infamous for the very poor quality of its waxworks, a quickfire buzzer round was played where the panellists had to guess who the waxworks were meant to be of. Appears to discuss various tiebreaker rounds with minimal emphasis on a "first to score a point wins" mechanic.
• Film.Roll Bounce: This example consists of a lot of dialogue where the characters agree to break a tie with a match where the winner takes it all
• Taskmaster.Tropes R To Z: If two or more contestants are tied in first place at the end of an episode, then a tiebreaker happens. Lists many examples, most of which don't really fit the strict definition.
• Tropes P to Z: Originally, nighttime ties were broken by a Speed Up round between the tied contestants. Since the introduction of Toss-Ups, a fourth Toss-Up is used instead. A tiebreaker Speed Up was also used on one episode of Wheel 2000, although without a Final Spin as that version was played for points. Some recollections mention one happening in the Goen era. Neither type of round appears to be SD.
• Series.Pyramid: Ties are broken by playing a round of words, with the team who created the tie getting a choice between two letters, which would be found at the beginning of each word in the tiebreaker round. As mentioned above, this got an Obvious Rule Patch. The 2012 version made it more straightforward: each team got 30 seconds, and kept going until the time was up. Whichever team got the higher score in the 30 seconds won. The 2016 version eliminated this, mostly. If there was a tie, the winner was determined by the total time the team got to the tying score. The first time it came around, the winning team won by 0.5 seconds.
• Series.They Think Its All Over: From Series 2 onward, if an episode finished level after "The Name Game", the teams would play a tiebreak game, the formats of which were often Call Backs to earlier rounds. If the two teams had won equal numbers of episodes by the end of a series, they would also play a tiebreak for the series. Examples of tiebreak games included answering questions from trivia books purportedly written by David Gower and Gary Lineker (neither captain could answer questions from "his" book), musical chairs, a race on "skeleton bobsleds" (tea trays with skateboard wheels), launching football boots at cutouts of David Beckham's face,^note  A reference to a then-recent incident in which Sir Alex Ferguson (supposedly accidentally) kicked a boot at Beckham during a dressing room tirade when Manchester United were trailing 2-0 at Arsenal. riding a mechanical bull, and seeing who of Rory and Jonathan could hold their breath for the longest while face down in a bowl of water.
• WebOriginal.Tapestry Of Horror: Happens in Round 2, Void vs Mortie, where the judges disputed about coming up with a conclusion. Both were asked to make an epilogue within a short time limit to help decide a winner.
• TabletopGame.Twenty Two: In the event the final players go bust at the same time, and there is a tie for lowest score, standard tie breaking procedure is to run one last hand between the tied players - winner takes all.
• Series.Finders Keepers: If the game ended in a tie after two rounds, a shortened hidden pictures round was played with the first to two correct answers moving to the Room-to-Room Romp.
• Mahjong: In the Japanese Riichi variant, some rules will have the game continue into a West round (or a South round in an East-only game) if no one has reached a quota (usually 30,000 points) by what would have been the end of the game. Some rules will end the game as soon as someone exceeds the quota, or sometimes the entire extra wind is played. This is just a race to reach 30,000 points first.

• VideoGame.Super Mario Bros 35: If the last few players drag the game on for too long, the timer will turn red and never stop counting down quickly. It initially counts down at quadruple speed, but should both players still manage to hang on for a long time, it will start counting down at x10 speed. This isn't even an extra round. It's just the game telling the players to hurry up.
• VideoGame.Worms: Rise to the Challenge: One of the most common effects during Sudden Death. The water will rise after each turn, slowly taking the map down and eliminating worms that weren't able to save themselves. Focuses on the game telling you to get it over with already, with no mention of any "first to score wins" mechanic.
• VideoGame.Metal Gear Online: Capture Mission: Now both teams have two capture points, meaning it is now possible to speed up the timer by capturing both KEROTAN and GA-KO, and taking them to both capture points. When the time expires, the game can enter into Sudden Death mode, in which whatever time is counted down no longer resets. Seems to be another case of just the game penalizing players for taking too long.
• TabletopGame.Fury Of Dracula: Dracula's titular "Fury of Dracula" ability. After three weeks have passed, he will gain influence on every (land) turn, thus putting huge pressure on the hunters who have to act fast or lose.
• VideoGame.Power Bomberman: When time is running out (default is when 50 seconds are left, but this can be customized), indestructible blocks will fall into the stage and crush any player below them. They also fill up the stage, making the playable area smaller. As per the rest of the series, this is also subverted, as a draw can still happen if the last few blocks knock out the remaining players all at once (or even within a second or so of each other).
• VideoGame.Play Station All Stars Battle Royale: Going into Overtime mode causes everyone to earn AP twice as fast. An interesting part of the game's Sudden Death mechanic is that all participating fighters (not just the ones who force Overtime) will participate in the final battle, allowing for come-from-behind victories. If the game is still tied after Overtime, the AP rate is multiplied by 3. It can eventually go up to 4. No mention of any "first to pull ahead wins" aspect. It just seems to make the game more volatile than usual.

• Characters.Diep Io Sudden Death: Their appearance heralds this rule being applied to the current game; any player that dies (not necessarily by an Arena Closer, but very likely) is booted out of the server. The tanks in the game don't fall under One-Hit-Point Wonder, so you can come back even if you get hit first. This also does not appear to be a tiebreaker.
• VideoGame.Fortnite: In normal game modes, if significantly more time than usual transpires within a match, the Storm Eye will just straight up vanish completely, at which point everyone will rapidly lose health from the Storm until only one person or team is remaining. Normally in Disco Domination, players respawn after death — but once the storm shrinks for the third and final time, they stop. If an entire team is eliminated, the other one wins instantly regardless of points. Contains one example similar to the Diep.Io one, and one instance of "just the game penalizing players for taking too long".
• Combat Resuscitation: Battle Royale restricts this to team-based modes without respawning, leading to odd cases in certain LTMs where being KO'd early in the match counts as an immediate elimination, but later in that same match you can crawl around for several seconds first because Sudden Death has begun and respawning is no longer an option. Problematic for similar reasons as the above
• Pinball.Rollergames: Large-Ham Announcer: Both the male and female announcers go wildly bombastic over every little event.
  Male Announcer: "It's Sudden Death! GO FOR THE WALL!"

After looking it up, this "sudden death" appears to be just a particularly impactful part of the game.

• Popular Game Variant: TabletopGame.Go has a Sudden Death variation where players start with two stones each placed in a square on the board, the goal being to capture an opponent's piece first to win. This has very little to do with the normal playstyle (which is about building up territory with the stones), but it is a lot faster. A strange one, considering that this is not how you "pull ahead" in the normal game. Looks valid except for the fact that this doesn't seem to be a tiebreaker

• TabletopGame.Fluxx: Averted. If two or more players accomplish the same goal (or either goal by the Double Agenda card) at once, the game continues until only one player is the active winner. This also works for Ungoal cards. It's unclear how this one interprets the trope, as it merely mentions that "keep playing until there's only one active winner" doesn't count.
• VideoGame.Supre Smesh Bras: A rare chance of both players losing their final stock at the same time results in a Saden Deth. The stage restarts No mention of what happens after the stage restarts.
• Wrestling.Backlash: In 2001, this happens in the 30-minute Ultimate Submission match between Chris Benoit and Kurt Angle, in which Benoit wins at 4-3. Tells you that Sudden Death took place, but says nothing about what it actually is.
• Golden Snitch: The "1-1-2" resolves both problems because now all rounds always matter — even a badly losing player coming from behind to win the third round will only tie the score, often leading to a Sudden Death round which can be really dramatic. Just a nod to Sudden Death's role as a tiebreaker
• VideoGame.Head Sports: Both Head Soccer and Head Basketball have a Sudden Death when the timer runs out, and Head Boxing has a "Double KO", which is exactly the same.
• Game Show Tropes: Since we have a tie score, We have to break the tie.... Just "Sudden Death is a form of tiebreaker."
• AwesomeMusic.Gundam: The first opening theme, "Cerulean" by BACK-ON, which really sets the tone of the show and accompanies Build Burning Gundam's reveal, as well as the Sudden Death round between Sekai and Wilfred in the tournament finals.
• YMMV.Gundam Build Fighters Try: The first opening theme, "Cerulean" by BACK-ON, which really sets the tone of the show and accompanies Build Burning Gundam's reveal, as well as the Sudden Death round between Sekai and Wilfred in the tournament finals.
• Recap.Total Pokemon Survivor Island S 1 E 20 Caught In The Crossfire: Venusaur and Gliscor go through this round upon being defeated by Reshiram's Fusion Flare at the same time, which Gliscor ends up winning. Tells you that SD happened, but not what it is
• Fanfic.Total Loud Island: Chris adds these to the game when the merge arrives in order to spice up the drama, resulting in several contestants being eliminated without any voting ceremony and sometimes much earlier than in canon. Suggests that the "sudden death" is not a "first to score a point wins" thing or even a tiebreaker, but just some way to eliminate contestants quickly.
• VideoGame.RWBY Amity Arena: After 3 minutes, the player with the most remaining buildings won; if the players still have the same number of structures, the match went into Sudden Death for 1 or 3 more minutes depending on the Arena. Never explains what the SD round actually entails, though the fact that it's timed makes me suspect that it's not a "the first to score wins" round (what if they're still tied after the extra round finishes?).