Is it possible to hit a golf ball on the surface of the moon and have it achieve a stable orbit around the moon?

We would need to work out the exit velocity required by gravitational forces.

If we tee'd it up on the highest point, the highest mountain of the moon, there might be a chance to achieve this, right ?

I think not.

Let R be the radius of the moon. It's about 1,700 km or 1.7 x 10^6 meters.

Let g be the acceleration of gravity at the surface : it's about 1.7 m/s^2.

We achieve circular orbit right at the surface if the acceleration of gravity exactly matches the acceleration needed to maintain uniform circular motion: that is, if

g = v^2/R or v = sqrt.(g*R).

This would be about

1700 meters per second. This is probably 40 or 50 times the speed any human golfer could swing a club.

N.B. 40 times the speed is 1600 times the kinetic energy.

Conclusion: It is not possible.