According to Newton's Second Law of Motion, if a net force acts on an object, the object accelerates acording to the equation:
Where m is the mass of the object.
Then, if an unbalanced force is acting on an object, the net force is different from 0, which means that the object will accelerate in the direction of the net force.
If an object has a uniform acceleration, it won't have a constant velocity, but it may have a constant speed anyway if the force is always perpendicular to the velocity, in that case the object would be moving in a circular trajectory.
In other words, an unbalanced force will cause an acceleration, this can either change the speed of an object or the direction of its velocity (or both).
Then, it might be possible for an object to have a constant speed under the influence of an unbalanced force if that object follows a circular trajectory.