In metric fixed point theory, you have Darbo (and Sadovski) theorems. They are generalizations of Schauder fixed point theorem, but instead of assuming that the set (or the mapping) is compact, one assumes that it maps sets into "more compact" sets (using some kind of a measure of noncompactness). (I don't have any reference with me at home now, but one classical source is Granas' and Dugundji's monograph on fixed point theory.) For instance, in Darbo's theorem, you assume that you have a continuous mapping $F$ on a non-empty bounded closed convex subset $C$ of a Banach space and a constant $k<1$ such that $\alpha(F(B))\le k\alpha(B)$ for each $B\subset C$, where $\alpha$ is the Kuratowski measure of noncompactness.
The notion of measure of noncompactness goes back to 1930s (Kuratowski was the one who introduced it, without relation to fixed point theory), Darbo's and Sadovski's theorems are (AFAIK) from 1950s (maybe a bit later).
On a related note, there are quite a few theorems (some fixed-point ones and some others) about hyperconvex metric spaces (a notion introduced in the 1930s by Aronszajn, but first appearing in a published paper in 1956) whose proofs employ simple tools from metric topology, but are highly nontrivial. A beautiful example is Baillon's theorems:
An intersection of a chain of bounded hyperconvex spaces is nonempty and hyperconvex.
Any nonexpansive (i.e., having Lipschitz constant $1$) mapping of a bounded hyperconvex space into itself has a fixed point.
Both can be proves using quite elementary methods (but rely heavily on the axiom of choice), but the proofs are far from trivial. Both were proved in a paper published in late 1980s.
(For more info on hyperconvex spaces, see e.g. http://en.wikipedia.org/wiki/Injective_metric_space and references therein, in particular the original paper of Aronszajn ana Panitchpakdi and the article by Espinola and Khamsi).