Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation.
1. **Understanding the Problem:**
   We need to find the maximal length of a non-constant arithmetic progression whose terms do not contain a specific digit \( k \) in their \( p \)-adic representation, where \( p \) is a prime number and \( p \geq 5 \).

2. **Common Difference and Units Digit:**
   Let \( d \) be the common difference of the arithmetic progression. If \( p \mid d \), the units' digit of the terms in the sequence never changes. This is because adding a multiple of \( p \) does not affect the units digit in base \( p \). Therefore, we can consider the sequence with common difference \( \frac{d}{p} \) instead, which still maintains the property of not containing the digit \( k \).

3. **Assumption on \( d \):**
   We can assume \( \gcd(d, p) = 1 \). This is because if \( d \) is not coprime with \( p \), we can reduce the problem to a smaller common difference as explained above.

4. **Invertibility of \( d \) modulo \( p \):**
   Since \( \gcd(d, p) = 1 \), \( d \) is invertible modulo \( p \). Let \( c \) be the modular inverse of \( d \) modulo \( p \), i.e., \( cd \equiv 1 \mod p \).

5. **Form of the Sequence:**
   Consider the arithmetic progression \( dn + e \). We need to ensure that none of the terms in this sequence contain the digit \( k \) in their \( p \)-adic representation.

6. **Modulo Condition:**
   For any term \( dn + e \) in the sequence, we have:
   \[
   dn + e \equiv k \mod p
   \]
   This implies:
   \[
   n \equiv c(k - e) \mod p
   \]
   where \( c \) is the inverse of \( d \) modulo \( p \).

7. **Range of \( n \):**
   Since \( n \) ranges over all integers, \( n \equiv c(k - e) \mod p \) will eventually cover all residues modulo \( p \). Therefore, one of the first \( p \) terms of the sequence \( dn + e \) will contain the digit \( k \).

8. **Maximal Length:**
   This shows that the length of the arithmetic progression without the digit \( k \) in their \( p \)-adic representation is at most \( p - 1 \). This length is achieved by the sequence \( 1, 2, \ldots, p - 1 \) itself, as none of these terms contain the digit \( k \) in their \( p \)-adic representation.

9. **Special Cases for \( p = 2 \) and \( p = 3 \):**
   For \( p = 2 \) and \( p = 3 \), the problem is trivial because the \( p \)-adic representation has very limited digits (0 and 1 for \( p = 2 \), and 0, 1, and 2 for \( p = 3 \)). Therefore, the maximal length of the sequence without a specific digit \( k \) is very small and does not provide meaningful insight.

\(\blacksquare\)

The final answer is \( \boxed{ p - 1 } \)