The Arabic word روح (rūḥ) means "soul", but also means "life", and they're used somewhat interchangeably in the Islamic literature. It makes sense to expect that only (some) living things can have souls. However, there's the caveat:
And they ask you, [O Muhammad], about the soul. Say, "The soul is of the affair of my Lord. And mankind have not been given of knowledge except a little."
Fatawa such as IslamWeb cite the hadith (concerning image making):
إن أبيت إلا أن تصنع فعليك بهذا الشجر ،كل شيء ليس فيه روح ...
... If you insist on making pictures I advise you to make pictures of trees and any other unanimated objects.
Sahih al-Bukhari 2225
and this is how they determine that plants don't have souls; see also Do plants have souls. This implies that some things don't have souls.
Another hadith (related to sports hunting) is:
لا تتخذوا شيئا فيه الروح غرضا ...
Do not make anything having life as a target
Sahih Muslim 1957 a
Here the Prophet forbade sport hunting for things that have a روح ("soul"). It doesn't make much sense to say this if everything has one.