What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors?

[i]Ray Li[/i]
1. We need to find the smallest perfect square larger than $1$ that has a perfect square number of positive integer factors.
2. Let's list the perfect squares and count their factors:
   - $4 = 2^2$: The factors are $1, 2, 4$. There are $3$ factors.
   - $9 = 3^2$: The factors are $1, 3, 9$. There are $3$ factors.
   - $16 = 2^4$: The factors are $1, 2, 4, 8, 16$. There are $5$ factors.
   - $25 = 5^2$: The factors are $1, 5, 25$. There are $3$ factors.
   - $36 = 6^2$: The factors are $1, 2, 3, 6, 9, 12, 18, 36$. There are $9$ factors.
3. We observe that $36$ is the smallest perfect square larger than $1$ that has a perfect square number of factors ($9$ factors, which is $3^2$).

Thus, the smallest perfect square larger than $1$ with a perfect square number of positive integer factors is $36$.

The final answer is $\boxed{36}$