Let's reexamine the question and my response to identify where the misunderstanding may have occurred.

Revisiting the Question
The question asks if it's possible, in a group of numbers where not all numbers are identical, for there to be no number smaller than the median.

Analyzing the Initial Explanation
In my previous response, I stated that the median divides the numbers into two halves, where at least half of the numbers are equal to or less than the median. This explanation assumes a standard scenario where the numbers are distinct.

Identifying the Mistake
Upon further reflection, the key lies in considering the presence of multiple identical numbers in the dataset. If the set contains multiple instances of the same number, and this number happens to be the median, it's possible that no number is smaller than the median. For example, in a dataset like [4, 4, 4, 4, 5, 6, 7], the median is 4, but there is no number smaller than 4 in this dataset.

Corrected Solution By Steps

Step 1: Reconsidering the Definition of Median with Identical Numbers
In a dataset with identical numbers, the median is still the middle value when the dataset is ordered. If the median happens to be the most frequently occurring number (mode), it is possible that all numbers in one half of the dataset are equal to the median.

Step 2: Applying this to the Given Scenario
In a scenario where the dataset consists mostly or entirely of the same number, which is also the median, there can indeed be no number smaller than the median. This is a special case that occurs when the median is equal to the smallest number in the dataset, especially in datasets with high repetition of a single number.

Final Answer
Yes, it is possible in a special case for there to be no number smaller than the median in a group of numbers where not all are identical, particularly when the dataset contains multiple instances of the median value, and this value is equal to the smallest number in the set.