geshem

What's the connection between the Hebrew word geshem - גשם - "rain" and gashmi גשמי - "physical"?

From my initial research there is none. Geshem is a biblical word for rain, and appears about as frequently in the Bible as its synonym matar מטר. In Talmudic Hebrew, however, geshem became the nearly exclusive word for rain, and so it is also today.

Gashmi was borrowed into Medieval Hebrew from Arabic, which in turn is cognate with the Aramaic geshem (or gishma גשמא) meaning "body." That word is also biblical, appearing a few times in the Aramaic section of the book of Daniel. From "body" it came to mean "substance, matter", and this also led to the verbs higshim הגשים - "was carried out" or "embodied" and hitgashem התגשם - "was realized, fulfilled."

Once these verb forms entered Hebrew, it became must less common to use the root גשם to refer to the act of raining (even though there are verbs like that in Biblical Hebrew), but rather the verb form of matar: himtir המטיר -  "to make it rain." From this root we also have the words mitria מטריה - "umbrella" and mamtera ממטרה - "sprinkler."

Many sources I found, including this one from the Academy of the Hebrew Language, said there was no connection between the two homonyms. However, there are those that claim that matar referred to any kind of rain, whereas geshem was a particularly heavy rain. According to this school of thought, geshem could be related to the Arabic jasuma, "to be bulky, thick", which would lead to a connection with the Aramaic geshem - "body" as well.