The word for a medical "mask" is borrowed from the English language. Does this infer that these types of masks did not exist in Japan before the arrival of the English? If these masks did exist in Japan in times of old, what was the native Japanese word for such a mask?
According to this article linked below, masks were introduced to Japan in the beginning of the Meiji period (1869) with coal miners to avoid inhaling harmful particles. At that time they were called 呼吸器(こきゅうき), essentially respirator. However they were not widespread until the Spanish Flu in 1918. In this picture you can see and advertisement for masks warning that not wearing a mask is reckless (not much has changed). According to this picture it seems that modern masks came along with their loaner word.
There are of course more words for mask in Japanese. The most common probably being 仮面, which is usually in the case of masks used for disguise or to alter ones appearance.