As @Robert Israel pointed out, by the definition of hyperbolic fixed point the number of fixed points is finite (or countable). 

A natural generalization of hyperbolicity for non isolated equilibria is that of [normally hyperbolic invariant manifold][1]. Several results that are known for hyperbolic fixed points carry to normally hyperbolic invariant manifolds,  in special when the invariant manifold is compact.

An excellent book about this matter - which also treats the case when the invariant manifolds is non-compact - is *Normally hyperbolic invariant manifolds — the noncompact case* by J. Eldering. The pre-proof version is [freely available online][2].

[1]: https://en.wikipedia.org/wiki/Normally_hyperbolic_invariant_manifold
[2]: http://wwwf.imperial.ac.uk/~jelderin/research.php