the first element of "gymkhana" comes from "gend" meaning "ball" in hindi/hindustani/khariboli. this element is distinct from english word "gym", short for "gymnasium" and "gymnastics" which has greek and latin roots. the second element, "khānā" (ख़ाना) has indo-persian or indo-arabic origin meaning a place or a compartment. in the persian (خانه) it's a term for "dwelling, house".