Is it possible for SOME positive $c$, $c<1$ to find a pair of COMPACT hyperbolic manfiolds $M^3$ and $N^3$ with a positive degree map $$f: M^3 \to N^3,$$ such that $f$ is contacting with constant $c$? Are there may examples like this?

One can ask the same question of Riemann surfaces, and it seems to me that this should be possible. For example we can take a double cover of Riemann surface with many points or ramification. Though I don't know a proof even in this case. Of course for non-ramified cover the best possible constant $c$ is $1$.

ADDED. Following the answer of Sam Need, let me give an approximative "proof" of the fact that this works in dimesnion 2. Let us triangulate a hyperbolic surface $N^2$ in triangles of very small size, that have acute angles (this is always possible). We want to show that a double cover of $N^2$ with ramifications at vertices of the triangulation will do the job. For this we need a lemma (without a proof).

Lemma. Suppose we have two hyperbolic trianlges, one very small and acute with angles $a$, $b$, $c$, and the over with angles a/2, b/2, c/2. Then there is a contacting map from the second triangle to the first one. The lemma is true, since the second trianlge will be large.

Now on the double cover we can take a trangulation that comes from $N^2$ and glue it from these triangles with half angles. Half angles come from doble cover. Then we just need to "adjust" the map.

Of course this is not a real proof, but I am 100% it can be made real.