The Big Ghost is a large, round white specter with black eyes and a pink mouth that drifts through the GLOOM in PEAK. You cannot fight it with normal items, but the stone Belltowers scattered through the biome double as a trap. Ring one at the right time and the ghost burns away in purple flame, which is exactly what the Exorcist Badge asks for.
Quick answer: Stand next to a Belltower in the GLOOM, wait for the Big Ghost to float up to its base, then hold E to ring the bell. The ghost is engulfed in purple fire and disintegrates, unlocking the Exorcist Badge.

Where the Big Ghost spawns
The Big Ghost appears in the GLOOM, the foggy purple-hued area marked by tall stone Belltowers. It always spawns near these towers, so you do not need to hunt across the whole biome. Move to a Belltower and the ghost will drift into range on its own.
Because the towers are the only tool that can burn the ghost, your plan is simple. Pick a Belltower, get close enough to use it, and let the ghost come to you rather than chasing it.
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E until the yellow circular progress bar around the RING prompt fills completely. Timing matters, so start ringing just as the ghost closes in.
How to confirm the Exorcist Badge unlocked
You get two clear signals that the burn worked. The ghost disintegrates into purple particles instead of floating away, and the top of the Belltower lights up with a bright purple flame. Once the ghost is gone and the tower is lit, the Exorcist Badge is credited.
The most common reason it fails is timing. If you ring the bell too early or too late, the ghost is not at the base, and the flames never catch it. Wait until it is right at the tower before completing the ring.
Tips for surviving the GLOOM while you wait
The GLOOM applies a sleep effect that can knock you out while you loiter near a tower. Bring a Candlestick and ring the Belltowers to hold off the sleep effect so you stay awake long enough for the ghost to arrive.
Note: playing on TENDERFOOT difficulty makes the wait far easier, since hunger and stamina drain more slowly and there is no time pressure. That gives you room to reposition and re-attempt the ring if the first try misses.
Once you have the pattern down, the Exorcist Badge is one of the more repeatable objectives in PEAK. Find a Belltower, keep yourself awake, and ring the bell the instant the Big Ghost drifts into its base. The purple flame does the rest.






