As Robert Israel has 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. Several results that are known for hyperbolic fixed points carry over to normally hyperbolic invariant manifolds, in particular 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.
