The greatest five-digit number which is a perfect square is:

The greatest five-digit number is $99999$. We find its square root using the long division method to find the remainder.
Evaluating the square root of $99999$:

$$3\times 3=9$$

The next period is $99$. The divisor becomes $3+3=6$.

$$61\times 1=61$$

The remainder is $99-61=38$.
The next period is $99$, making the number $3899$. The divisor becomes $61+1=62$.

$$626\times 6=3756$$

The remainder is $3899-3756=143$.

Subtracting this remainder from the greatest five-digit number:

$$99999-143=99856$$

Thus, $99856$ is the greatest five-digit perfect square, which is equal to $316^2$.