Technically, they are playing roque, an American variant* of croquet, which is played on a hard sand or clay surface. Introduced in the late 1880s, it was extremely popular in the first few decades of the 20th century — and an Olympic sport in 1904 — and then almost entirely disappeared after the 1950s.
The photo above was published between 1900 and 1919, photographer unknown. It is part of the Termaine Arkley Croquet Collection and via UBC (University of British Columbia) Library Digitization Centre Commons on flickr.
There’s also a photo of 1918 roque grounds in Florida here.
*There is also a modern game of beach croquet.