Jelly, 17 bytes

FAṀŒRṗLƲṁ€ðæ*2iị⁸

Try it online!

It's been over a week, so I figured I'd reveal my brute-force Jelly approach. This makes the following assumption, which I haven't managed to prove, but can't find a counter example:

Let \$B\$ be the input \$n \times n\$ matrix, and \$A\$ be an \$n \times n\$ matrix such that \$A^2 = B\$

Let \$b = \max\{|B_{ij}|, 1 \le i,j \le n\}\$ i.e. the largest absolute element of \$B\$. This answer assumes that, assuming \$A\$ exists, there is at least one matrix \$A\$ such that all its elements are in the inclusive range \$[-b, b]\$

For example, for $$B = \left[ \begin{matrix} 9 & -8 \\ 0 & 1 \end{matrix} \right]$$,$$B = \left[ \begin{matrix} 9 & -8 \\ 0 & 1 \end{matrix} \right],$$ \$b = 9\$ and there exists at least one \$A\$ such that all elements are between \$-9\$ and \$9\$ (the example in the question, for example)

How it works

FAṀŒRṗLƲṁ€ðæ*2iị⁸ - Main link. Takes B on the left
F                 - Flatten B
 A                - Absolute values of each.
       Ʋ          - Last 4 links as a monad f(abs(flat(B)):
  Ṁ               -   Maximum
   ŒR             -   Bounced range; [-max, +max]
      L           -   Length i.e. number of elements of B
     ṗ            -   Powerset
         €        - Over each list:
        ṁ         -   Mold it like B
                    Call this list of matrices M
          ð       - Begin a new dyadic chain with M on the left and B on the right
           æ*2    - Matrix square of each matrix in M
              i   - Index of B in M
                ⁸ - Yield M
               ị  - Index back into M

One byte longer is a slightly more conventional way of extracting the root:

FAṀŒRṗLƲṁ€æ*2⁼ɗƇ⁸Ḣ

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Here, the dyadic filter æ*2⁼ɗƇ means that it would save a byte over the dyadic chaining with the indexing. However, as we should only output one solution, and the Ḣ would chain to the filter without either ⁸ or ¹, the indexing here actually saves a byte.