$22$ football players took part in the football training. They were divided into teams of equal size for each game ($11:11$). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training. 
1. **Define the problem and notation:**
   Let \( f(n) \) be the minimum number of games required for \( n \) players such that each player has played against every other player at least once. In this problem, we have \( n = 22 \) players, and they are divided into two teams of 11 players each for each game.

2. **Initial observation:**
   Each game involves two teams of 11 players each, so each player in one team plays against all 11 players in the opposing team. Therefore, each game allows each player to play against 11 new opponents.

3. **Recursive relationship:**
   To ensure that each player has played against every other player at least once, we need to consider the recursive nature of the problem. If we denote the number of games required for \( n \) players as \( f(n) \), we can express it as:
   \[
   f(n) = 1 + f(\lceil \frac{n}{2} \rceil)
   \]
   This is because, after the first game, we need to ensure that the remaining players (half of the original set) also play against each other.

4. **Base case:**
   For \( n = 11 \), we need to find the minimum number of games such that each player has played against every other player at least once. This can be represented as:
   \[
   f(11) = \lceil \log_2(11) \rceil
   \]
   Since \( \log_2(11) \approx 3.459 \), we have:
   \[
   \lceil \log_2(11) \rceil = 4
   \]

5. **Calculate for \( n = 22 \):**
   Using the recursive relationship, we have:
   \[
   f(22) = 1 + f(11)
   \]
   Substituting \( f(11) = 4 \), we get:
   \[
   f(22) = 1 + 4 = 5
   \]

6. **Conclusion:**
   Therefore, the smallest possible number of games required for 22 players to ensure that each player has played against every other player at least once is 5.

The final answer is \(\boxed{5}\).