WASHINGTON—The immigration-court backlog grew by at least 10% during the partial government shutdown, as a funding dispute centered on border security left the nation’s overloaded immigration system digging out of an even deeper hole than before the five-week standoff.

The court backlog, at more than 800,000 cases before the funding gap that started Dec. 21, swelled by at least 80,000 as courts stopped hearing most cases and Justice Department lawyers were furloughed, stalling litigation over immigrant-family separations and...