How can I have a permanent status line showing the name of the current file?

From the manual, you're really just wanting:

set laststatus=2
set statusline=%f "tail of the filename

As an aside: if you're wanting a status line with a ton more information be sure to check out this link. It'll give you something a bit more fancier like so:-

You can use Ctrl+g to print the current file name along with the cursor position and the file status.

To permanently show the current filename (and much more) in the statusline, you can use the vim-airline plugin. By default, the statusline will look similar to this: