The answer is "no" because only for a black hole can light never escape.
However, this question deserves a fuller answer because of a thing called the "photon sphere".
At the Schwartzchild radius of a compact body - whose mass is all inside the radius - light can no longer escape. However, at a distance which is 1.5 times the Schwartzchild radius there is a thing called the photon sphere. Any light incident on a massive body that comes from the outside of the photon sphere and enters it will invariably be "sucked into" the body. Light that originates within the photon sphere, but outside the Schwartzchild radius, and is directed outwards, may still escape however. This opens up the possibility that there exist objects so massive that their mass is within a photon sphere but not within the Schwartzchild radius. Such objects would likely be neutron stars.
Now the fun part - what do we mean by "invisible"? If by "invisible" we mean "I shine light on it and zero of it is ever reflected" then the answer to your question is "yes"!