The smallest positive integer that does not divide $1\times 2\times 3\times 4\times 5\times 6\times 7\times 8\times 9$ is:

Let $P=9!=1\times 2\times 3\times 4\times 5\times 6\times 7\times 8\times 9$.
The prime factorization of $9!$ is:

We test positive integers in increasing order starting from 1:

• All integers from 1 to 10 only have prime factors 2, 3, 5, and 7, and their exponents are smaller than or equal to those in $9!$. For example, $10=2\times 5$, which divides $9!$.

• The next integer is 11, which is a prime. Since 11 does not appear in the prime factorization of $9!$, it cannot divide $9!$.

Therefore, the smallest positive integer that does not divide $9!$ is $11$.