Structures for a nanosheet field-effect transistor and methods for forming a structure for a nanosheet field-effect transistor. A fin is formed that includes a first nanosheet channel layer and a second nanosheet channel layer arranged in a vertical stack. A cavity is formed between a portion of the first nanosheet channel layer and a portion of the second nanosheet channel layer. An epitaxially-grown source/drain region is connected with the portion of the first nanosheet channel layer and the portion of the second nanosheet channel layer. A gate structure is formed that includes a section located in a space between the first nanosheet channel layer and the second nanosheet channel layer. The cavity is surrounded by the first nanosheet channel layer, the second nanosheet channel layer, the section of the gate structure, and the source/drain region to define an air gap spacer.