A more Workplace-y answer but also usable as an IPS:
Wherever you go, you will always have a detractor. It may be someone that feels threatened by you. It may be someone who just doesn't like you for whatever reason. It may be someone who needs to expand their position and feels the need to do it at your expense. Or it may be someone who is just a jerk. Either way, you wind up with rumors and gossip to contend with. I've seen this in close to 40 years of working at almost every place I've worked.
At work, there is one person that you need to impress: your boss. The others are, although not irrelevant, secondary. Your boss is the one that can fire you and determines your career advancement. Co-workers come and go.
If this gossip concerns you, sit down with your boss. Explain what's going on and your concern. Get some honest performance feedback on how you are doing, and address that. Don't get defensive; merely explain that you've heard rumors and you don't want that to affect your relationship with your boss. Don't try to prove that wrong; that merely sends a red flag. And keep in contact with your boss. You don't need to be friends, but you do want your boss to see you as someone that they can depend on and to deliver consistent quality results.
Now, with respect to your friends (and not always at the workplace): don't get defensive here. My response is usually something like "[X] can flap their gums all they want. I let my results speak for themselves" and let it go at that. That noise may bother you, but you can't let it appear to do so. Merely stay confident, and deliver consistently. When people see that you get stuff done and done well (including non-work activities), your co-worker will make him/herself look small by spreading (known to be) false gossip. That shows you to be the grown-up in that interaction, and by playing the long game, you don't get down to that level.