I saw this yesterday in a company’s announcement about buying another company.

Problem:
The hyphen does not belong in this phrase.

Explanation:
One company was announcing the pending purchase of another company.

The purchase required approvals by various U.S. federal agencies.

The buyer wanted investors and employees to know that it was legally bound to continue to operate distinctly from the other company until all of those approvals had been secured.

The complete sentence announcing this legal constraint was along the lines of “We will continue to operate as a separate business at arm’s-length from [the company that we are buying].”

The problem with this sentence is that the hyphen does not belong.

The possessive “arm’s” is modifying the noun “length”, but together they are not modifying anything else, so no hyphen should appear between “arm’s” and “length”.

In contrast, a hyphen does belong in a phrase such as “arm’s-length transaction”.

Solution:
“… separate business at arm’s length from …”