I think I can find a cover that has no finite subcover. Consider open n-balls such that their size decreases as they approach the boundary of $D(0,1)$ Poincaré Hyperbolic Disk fashion. Then this cover is going to have no finite subcover since you cannot take any subcollection of it that is finite and still covers all of $D(0,1)$, making it noncompact
I am not sure how to formulate that though, I obviously need a sequence that decreases to zero for the radii of each n-ball in the cover, but the extra degrees of freedom in higher dimensions $dim X > 2$ makes it difficult to specify the direc…