Basilisk is a free, open-source desktop web browser built on the Unified XUL Platform (UXP). It is an updated fork of Firefox that keeps the look and feel of Firefox versions before 57 and uses the Goanna rendering engine, which it shares with Pale Moon. UXP emphasises maintainability and keeping features and extensions usable over long periods. Basilisk runs on Windows and Linux, with experimental support for FreeBSD and macOS. It is installed locally as an application; there is no hosted service behind it.
Basilisk is maintained independently by the Basilisk Browser Development Team and is not affiliated with Mozilla. According to Wikipedia, the source code is under the MPL-2.0 licence, while the official binaries are distributed as proprietary freeware. Because the browser runs on your device, no vendor-operated cloud holds your data, but you remain responsible for keeping it updated. The project does not state where its maintainers are located, so for a European organisation the governance of the project and its update practice are the points to assess.