As of Wuthering Waves Version 1.0, characters who are using Lanternberry as ascension materials are Jianxin and Baizhi. You can find this ascension mat around Tiger's Maw Mine in Tiger's Maw.
All 43 Lanternberry in Wuthering Waves
You can find the first 28 Lanternberry in Tiger's Maw Mine. The circled icon in the image above shows 1x of Lanternberry, unless stated otherwise (x2 or x3). Most of these berries are on the vine on the rock or the wall.
There are some Lanternberry on the surface of the mine, while some are inside the mine. It may be confusing since the map will not show the underground area (unlike Genshin) so please refer to the video above to see the location of each berry on the map.
For the last 15, you can buy it from Koko in Shifang Pharmacy, Jinzhou. You'll need a total of 45,000 Shell Credits to buy all of them.