Edited after reading Mike Walsh's post:
If you're looking for a web-browser which is not based on firefox and has little to do with chromium project see this post, viewtopic.php?p=16903#p16903. It's relationship to chromium is explained here, https://wiki.qt.io/QtWebEngine/Rebase_on_New_Chromium and https://doc.qt.io/qt-5/qtwebengine-overview.html
Although that post pertains to Bionicpup64, https://pkgs.org/search/?q=falkon reveals that versions are available in Focal Fossa, Slackware and debian repositories, even some 32-bit versions. However, unless your Puppies Package Manager has access to a repo containing it, or you are familiar with pkg-cli, http://www.murga-linux.com/puppy/viewtopic.php?t=112927 obtaining a functioning version will be an up-hill battle. The Bionicpup64 version had 96 dependencies --all of which, however, PPM automatically downloaded and PaDS, viewtopic.php?p=6355#p6355 combined, in a couple of minutes, into a 52 Mb SFS which played Youtube Music OOTB.