The first counter-example is for $p=17$. The interval of length $40$ starting at $87890$ only yields three integers with least prime factor greater than $17$: the primes $87911,87917$ and also $87929=23\cdot 3823$. So we can use the inteval of length $38$ starting at $87890$ or at $87891$.

I may be mistaken, but I think that there are not any examples for $p=19$ and $p=23$. 

On the other hand, of the 89 consecutive integers from $43559563512434$ to $43559563512522$ inclusive, all but **one** have a prime factor $41$ or less. The exception is $43559563512481=9393910613\cdot 4637$


There are essentially 6 ways to get a run of length $39$ (or 3 up to reflection): One can start at $87890,177980,182342,328130,332492$ or $422582 \mod 510510$.