[BillyMac]

We have a few situations where team members are not required to "sit a tick", and some don't even involve playing with only five players.

3-3-1: Each team consists of five players, one of whom is the captain. Note: A team must begin the game with five players, but if it has no substitutes to replace disqualified or injured players, it must continue with fewer than five. When there is only one player participating for a team, the team must forfeit the game, unless the referee believes that team has an opportunity to win the game.

8.2 Situation B: A1 is fouled and will be shooting two free throws. After A1’s first free-throw attempt, B6 (Team B’s only remaining eligible substitute) replaces B2. A1’s second free-throw attempt is unsuccessful. During rebounding action for A1’s missed second free-throw attempt, and before the clock starts, A1 pushes B3 in the back causing B3 to roll an ankle. Team B is in the bonus. B3 is unable to immediately continue playing. Team B requests and is granted a time out in order to allow B3 to recover from the ankle injury so as to remain in the game. B3 is still not able to play after the time out has ended. Ruling: B2 may return to the game and replace B3 and shoot B3’s free throw attempts despite having been replaced since he/she is the only available substitute. (3-3-4)

3.1.1 Situation: After six players have been disqualified, Team A has only four who are eligible to continue in the game as players. In a gesture of fair play, the coach of Team B indicates a desire to withdraw a player so that each team will have four players on the court. Ruling: This is not permissible. Team B must have five players participating as long as it has that number available. If no substitute is available, a team must continue with fewer than five players. When only one player remains to participate, that team shall forfeit the game unless the referee believes this team still has an opportunity to win the game.

2002-03 NFHS Basketball Rules Interpretations
Situation 5: Team A is playing with five players, but has no remaining substitutes available when one of the players has an asthma attack. The coach is beckoned onto the floor. Ruling: The player must leave the game unless a time-out is requested and granted to Team A with the player being ready to resume by the end of the time-out. The team may continue with fewer than five players if there are no substitutes available. An injured/ill player may return to the game after recovery. (3-3-5)

However:

2018-19 IAABO Refresher Exam
41) Team A has no timeouts remaining and has only five players and no substitutes. A-2 appears to be injured and the coach enters the court to attend to A-2. A-2 is OK. The official informs the coach that A-2 must stay in the game and charges Team A with an excess timeout. Is the official correct? Original IAABO answer was “Yes”, later deemed a mistake and changed to “No”.

Also:

2000-2001 NFHS Basketball Rules Interpretations
Situation 5: A1 is fouled in the act of shooting and is awarded two free throws. A1 is injured on the play and Team A’s coach has been beckoned on the floor to attend to A1. Although by rule A1 should leave the game until the next legal opportunity to substitute, the officials make an error and permit A1 to complete the free throws before leaving. The opposing coach objects and insists that A1’s substitute should have shot the free throws, and a correctable error has occurred. Ruling: The officials made an error by permitting A1 to remain in the game. This is not a correctable error for a wrong player attempting a free throw, even though A1 is required to leave the game by rule. (3-3-5; 3-3-3)

2005-06 NFHS Basketball Rules Interpretations
Situation 1: A1 is fouled by B1 late in the second quarter. It is a common foul and the seventh Team B foul. The bonus situation is not recognized by the scorer or the officiating crew, and the Team A coach substitutes A6 for A1. A6 is beckoned onto the floor and A1 goes to the team bench. The scorer recognizes the error and sounds the horn (a) just before or (b) just after the administering official hands the ball to A2 for a throw-in. Ruling: This is a correctable-error situation and falls within the proper timeframe for a correction. In both (a) and (b), A6 leaves the game with A1 re-entering to shoot the bonus free throw. Play is resumed as after any free-throw attempt(s). If the second free throw is successful and the coach desires, A6 may re-enter the contest. (2-10-1a; 2-10-6)

8-2: The free throw(s) awarded because of a personal foul must be attempted by the offended player. If such player must withdraw because of an injury or disqualification, his/her substitute must attempt the throw(s) unless no substitute is available, in which case any teammate may attempt the throw(s) as selected by the team captain or head coach.