This is actually a really big problem for me as I find it physically discomforting to listen to lies being spoken unopposed.
Diplomacy provides for ample indirection: I'm so sorry to inform you so directly, I mean no offense, but what you're saying is completely unprovable, you have no facts at all that can demonstrate your position. I do appreciate your bravery for being willing to expose your ignorance on this topic so publicly, however.
Et cetera.
Such encounters are a rate opportunity to gain first-hand insight on someone else's particular set of unbearable truths. Maybe start with "What whole-Earth map would you recommend?"
I feel like the earth being round is such a generally accepted fact that anyone still clinging to flat-eartherism is either a troll or quite close to the extreme of that spectrum and I don't think it's worth my time to deal with those people - and I get that, it's sort of a mean stance to take but... I don't know, there's enough stuff I have to do in life - I'd rather not devote the effort into trying to convince someone who might just be trolling for the hell of it.
I have a "flat earther" friend on Facebook. I didn't know that about him until after I'd already friended him, but he posts about it constantly. After an initial private message of "Are you for real?", I've made it a point to as mildly as possible point out the obvious logical fallacies in the things he posts. I don't expect I'll ever change his mind - it's possible, but you can't reason someone out of a position that they didn't reason themselves into. Instead, I'd just like to have him see inconsistencies in what others who believe what he believes say. Over time I'm hoping that will at the very least keep it from spreading.
I'm living proof that this isn't true. I was a Young Earth Creationist for nearly 30 years, despite having an engineering degree from one of the world's best universities. I no longer believe in anything supernatural. It took a very long time, but I was eventually reasoned out of a position I wasn't reasoned into.
When people incorporate ideology into their identity, challenging the ideology directly is often interpreted as a personal attack or something else not worth honestly engaging with. A far better approach is to examine the methods people use to reach conclusions. There's a conversation technique called Street Epistemology that does exactly that. IMHO, it's the most productive way to have these kind of conversions.