In short, the answer is no. As you approach the speed of light, your kinetic energy will increase without limit. At the speed of light, kinetic energy would by infinite. Given the principle of conservation of energy, you need to account for that arbitrarily large increase of energy: where does it come from? When you use gravity to accelerate an object, the increase in kinetic energy comes to the expense of the energy of the body exerting the gravitational force.

This, by the way, is what space probes do when they use Jupiter (for example) to increase their speed without burning an extra amount of fuel (please google "Gravity assist"). The energy gained by the probe is given up by Jupiter.