You claim it isn't possible. I claim I can come up with a simple counter example to what you just stated. Try to come up with a simple counter example to your statement on your own.

Chat GPT4:
You are correct to challenge my initial response. Upon reconsideration, there can indeed be a counterexample to my initial claim. Let's consider a simple set of numbers where not all are identical, yet there is no number smaller than the median:

Suppose we have a set of numbers: [4, 4, 4, 5]

In this set, the numbers are not all identical (there's a 5 which is different from the 4s).
The median in this case, for an even number of observations, is the average of the two middle numbers. Here, the two middle numbers are both 4, so the median is 4.
In this set, there is no number smaller than 4.
Hence, this serves as a counterexample where not all numbers are identical, yet there is no number smaller than the median.