Find the smallest multiple of 9 that does not have any odd digits.
As the number is divisible by 9, the sum of the digits is divisible by 9.

On the other hand, since all the digits are even, the sum of the digits is also even. Therefore, the sum of the digits is at least 18. The smallest multiple of 9 with the sum of the digits equal to 18 is 99, but its digits are odd. This implies that the number must have three or more digits.

If we want the smallest number with 3 digits, the first digit must be at least 2. In this case, the sum of the other two digits is equal to 16 and since they are even, the only possibility is 288.

Therefore, $288=9 \times 32$ is the smallest multiple of 9 with all even digits.