I've got two (very similar) approaches to this. There's no way to do it with the built in write command, but it's fairly easy to generate your own function that should do what you want (and you can call it what you like - even W if you want).
A very simple approach that will only handle single-line ranges is to have a function like this:
command! -nargs=1 -complete=file -range WriteLinePart <line1>,<line2>call WriteLinePart(<f-args>)
function! WriteLinePart(filename) range
" Get the start and end of the ranges
let RangeStart = getpos("'<")
let RangeEnd = getpos("'>")
" Result is [bufnum, lnum, col, off]
" Check both the start and end are on the same line
if RangeStart[1] == RangeEnd[1]
" Get the whole line
let WholeLine = getline(RangeStart[1])
" Extract the relevant part and put it in a list
let PartLine = [WholeLine[RangeStart[2]-1:RangeEnd[2]-1]]
" Write to the requested file
call writefile(PartLine, a:filename)
endif
endfunction
This is called with :'<,'>WriteLinePart test.txt
.
If you want to support multiple line ranges, you could either expand this to include varying conditions, or you could pinch the code from my answer to this question. Get rid of the bit about substituting backslashes and you could then have a very simple function that does something like (untested though...) this:
command! -nargs=1 -complete=file -range WriteLinePart <line1>,<line2>call writelines([GetVisualRange()], a:filename)