This is definitely a legitimate concern and one that can't be completely avoided but it can be mitigated.
The first thing to consider is whether people rank at the bottom because the standards are too high or because they are not a good fit for their position. This is where relative performance comes in. Being at the bottom of the leaderboard isn't so bad if you're not far from everyone else. If the gap between last place and second to last place is huge then there may be a problem (for the person at the bottom).
The second thing to consider is that it's not necessarily bad to identify poor performers. It makes it easier for managers to identify who they should work with and helps employees know they need to step things up.
With those things in mind, if gamification is causing anxiety here is what I would recommend. You'll need to shift the focus so that it is more about recognition that performance evaluation. What I mean by that is that achievements (for example) should be viewed as a celebration of good work rather than an expectation of normal work. Help people understand the expectations. Consider building a more cooperative system where the gamification is on a team level rather than an individual level. You may or may not also have competition among teams.
You're always going to have people who compare themselves to the best and that may create seemingly unattainable standards for some people. Gamification has the added benefit of exposing to a degree what those people do to be good and in a well designed system, their performance can act as a blueprint for others to follow.
Ultimately you have to understand your culture. A Customer Support team might thrive on cooperation and generally needs a different approach than a Direct Sales team that feeds on competition. The competitive nature of the system can be adjusted by focusing on different things and building systems that emphasize team goals rather than individual goals if needed.