Well the body will accelerate as long as you are applying a force on it. So if u apply a force for forever it will accelerate forever but there is an interesting thing that happens.

Force applied for ever

You would notice that if u apply a force for an infinite amount of time then by $F=ma$ there should be acceleration for an infinite amount of time leading to a speed greater than light. Then is the time when you start to think that there is something wrong with this and indeed there is. Turns out that $F=ma$ isn't very accurate in this case but does this violate newton's second law? No, because newton's second law was never $F=ma$. This is a form it gets reduced to if the mass remains constant but we know that mass does increase with increase in velocity and considerably when speed reaches the speed of light.(The second law FΔt = p is still valid). So now we can see that as you reach the speed of light u need a larger force to have the same acceleration and at speeds really close to speed of light u will need an infinite amount of force to produce any acceleration in the body.

I know you have already had your answer but thought I should share this case which I really found interesting.