Download this to a ~/.textadept/modules/file_browser/ folder.

Go to the command entry and type:

require('file_browser').init('/path/to/project')

Pressing spacebar on any entry activates it. (Directories are opened/closed and files are opened in a split view). Other keys are: 'p' and 'n' to navigate up or down by item, 'P' and 'N' to navigate up or down by level, and 'f' and 'b' to navigate within a directory by its first and last files.

Alternatively, you can add file_browser = require('file_browser') to your ~/.textadept/init.lua and use file_browser in menus, scripts, and key bindings. For example:

local file_browser = require('file_browser')
table.insert(textadept.menu.menubar[_L['File']], 3, {
  'Open Directory...', file_browser.init
})