BOCA CHICA VILLAGE, Texas—In autumn 2019, Celia Johnson began resisting efforts by billionaire Elon Musk’s SpaceX to buy two modest houses she owns near the company’s rocket-launch facility.

Then she discovered that a 1,600-gallon water tank had gone missing at one of her houses, a rental property. Ms. Johnson said she and her neighbors quickly concluded SpaceX workers were the culprits. SpaceX denied responsibility but reimbursed her, she said, as it did when she accused its workers of later breaking into the vacant rental house and sleeping there.

“SpaceX bullied us from the beginning,” she said. “SpaceX employees did what they wanted.”

SpaceX, the maker of rockets and spacecraft controlled by Mr. Musk, among the world’s richest people, is facing off against a handful of households that have refused to sell their properties in this remote unincorporated village. Some 30 small ranch houses sit on a sandy spit of land near the Mexican border and Gulf of Mexico beaches, where SpaceX rockets roar into the sky and then return for a landing. Some come crashing back on land.

Villagers said Mr. Musk’s company has tried multiple times to buy them out. Some took the money, and SpaceX used the homes for its workers. Holdouts, at least seven of them, said they want more from a billionaire who’s after their dream vacation homes.