It seems impossible, yet I'm thinking that maybe because the ball compresses against the bat a bit it acts a little like a spring, and DOES travel faster than the bat?
EDIT:
This is just a clarification, and not really part of the question, but I think it may be valuable. For people saying momentum is conserved, I'm not sure what you are imagining, but take a moment to think about the equation you keep on mentioning: $$M_\textrm{bat}V_\textrm{bat} = M_\textrm{ball}V_\textrm{ball}$$ This is saying that the bat somehow transfers ALL of its momentum to the ball.
The only way this can ever happen is if the bat comes to a dead stop when it hits the ball, somehow holds it in place while transferring ALL of its momentum to it (that phrase doesn't even make logical sense), and then the ball flies off at a much larger velocity.
WERE the bat floating through space and struck a ball, the bat would not stop when it hits the ball, and there is no way it makes sense for the ball to go off at THAT much of a larger velocity. Just imagine a spaceship very slowly drifting through space, and an astronaut who suddenly touches it. Conservation of momentum DOES NOT mean $$M_\textrm{ship}V_\textrm{ship} = M_\textrm{astronaut}V_\textrm{astronaut}$$
According to that, the astronaut would shoot off at hundreds (maybe thousands) of kilometers per hour when being touched by a massive spaceship, and the spaceship would come to a stop, but obviously that doesn't happen. TOTAL momentum is conserved, so that $$M_\textrm{ship1}V_\textrm{ship1} + M_\textrm{astronaut1}V_\textrm{astronaut1} = M_\textrm{ship2}V_\textrm{ship2} + M_\textrm{astronaut2}V_\textrm{astronaut2}$$
but even THAT equation doesn't even apply in the case of the baseball strike since there is a human being providing a force EVEN as the bat hits the ball.
I know this is ingrained deeply in the minds of physics students because we have conservation of momentum drilled into our heads as young students in introductory physics, but I encourage everyone to always think intuitively about physics scenarios before applying equations.
Anyways, I hope that was valuable. Cheers!
Best Answer
Yes. Consider throwing a ball at a bat which is held stationary: the ball is momentarily stationary but at all other times it is moving faster than the bat.
Now consider sweeping the bat towards an initially stationary ball: if the ball is not to stick to the bat, then it must be moving faster than it when it loses contact with it. (This case is identical to the one above with a different choice of reference frame of course.)
In neither of these cases have I taken proper account of conservation of momentum: the bat must change velocity slightly when it imparts momentum to the ball, so you can't hold it stationary or sweep it at a constant velocity in fact. But this change in velocity of the bat can be made as small as you like by making $m_\text{bat}/m_\text{ball}$ large enough so the argument remains true.
Why we can ignore the person holding the bat
In the comments there has been some discussion about whether the person holding the bat makes a substantial difference. They don't: they certainly can make a difference in detail and obviously are responsible for getting the bat into the right position, but their contribution to the change in velocity of the ball is small. To see this I'll take some numbers from this page (mentioned in the comments).
The ball has a mass of $m = 0.145\,\mathrm{kg}$ and its change in speed $\Delta v \approx 200\,\mathrm{mph}$ or $\Delta v \approx 90\,\mathrm{ms^{-1}}$. This means that the impulse delivered to the ball is
$$I\approx 13\,\mathrm{Ns}$$
Now, let's assume that the person holding the bat exerts a force equivalent to their whole mass on it (they can't do this for any length of time, and in fact they can't do it at all realistically, so this is a safe upper bound). If their mass is $100\,\mathrm{kg}$, then the force they are exerting is $100\,\mathrm{kg}\times 9.8\,\mathrm{ms^{-2}}\approx 981\,\mathrm{N}$. The ball is in contact with the bat for $7\times 10^{-4}\,\mathrm{s}$ ($0.7\,\mathrm{ms}$), so the impulse from the person holding the bat delivered during the collision is
$$\begin{align} I_h &\approx 981\,\mathrm{N}\times 7\times 10^{-4}\,\mathrm{s}\\ &\approx 0.7\,\mathrm{Ns} \end{align}$$
So, the impulse delivered by the human holding the bat, in the best case, is about 5% of total impulse: realistically it will be much less.
This does not show that the human does not affect things like the direction and detailed trajectory of the ball after it is hit: it does show that their contribution to the change in velocity of the ball happens almost entirely before the impact: their job is mostly accelerate the bat and get it into the right place.
It turns out that Dan Russell has a nice summary page, with references on how much the person holding the bat matters. The last two sentences from that page are:
He has a lot of other useful information on the physics of baseball.