La Plata, originally named Grenada, was founded in 1890 by the XIT Ranch interests when Deaf Smith County was organized. The town engaged in a heated contest with neighboring Ayr to be the county seat, an honor that Grenada won in a controversial election on October 3, 1890. Soon afterwards the newly elected county judge, J. R. Dean, changed the name of Grenada to La Plata on request of federal postal officials. A small frame courthouse was built of lumber hauled from Amarillo. As the town grew, it added a post office, a school, a county jail, a Presbyterian church, and eighteen residents. Businesses included a general store, a pharmacy, a saloon, a hotel, an implement house, a blacksmith shop, a livery stable, and a printing office which housed the county's first newspaper, the La Plata Star.
The weather hindered the town's development almost from the beginning. From 1891 to 1894 the area around La Plata suffered a drought, which made farming and ranching almost impossible. Also disaster struck in February 1897, when a blizzard resulted in below-freezing temperatures for twenty-one consecutive days. More than half of the town's populace was compelled to leave. Moreover, in 1899 the Pecos and Northern Texas Railway built through the southern part of Deaf Smith County and into New Mexico. On November 8 of that year the citizens of La Plata chose a new county seat, the new town of Blue Water (or Bluewater, now Hereford), on the railroad. Nine houses, the courthouse, and the jail were loaded onto wagons and moved to the new location. Today nothing remains of the abandoned townsite except a few graves in the cemetery on land that has been reclaimed for farming. One memento of La Plata's brief life is the original portable jail, now on display at the Deaf Smith County Museum in Hereford.