The PA28 POH has a graph that will show the landing distance required and I recall from memory, it includes variable for 5 - 10 kts tailwind so I'd use that, and then compare the results to the runway length available. It's the only way. 670 metres isn't a particularly long runway, and I know the PA28-161 and the PA28-181 floats much more than the PA28-140's, so I don't think I'd chance it with that tailwind - even if the POH suggested it was just about OK (which I don't think it will)
You can operate in both directions if required due wind etc.
don't do anything you would feel uncomfortable with ... you are the pilot in command and can select / ask for the proper landing direction. If anything goes wrong with too much tailwind the judge will ask you exactly that ...
The Warrior POH has a 5 knot tailwind data point for LDR calculation. Your AFM may state a tailwind limitation, some do. If you use CAA safety factors (tailwind component taken at 150%, and 1.43x landing safety factor), and assuming airport is 1500' AMSL, on a summer day, dry tarmac, no slope the LDR over 50 foot obstacle is coming out at 785 metres. It goes without saying your insurance adjuster can carry out this calculation.
This assumes a perfect performance approach at Vat of 63 KIAS and 40o flap.
Note that on a go around the wind gradient would not be your friend.
Have you thought about a glide (or at least low powered) approach to land into wind? That's quiet.
I wonder why you bother land with a tailwind? Noise reduction procedures are not compulsory when safety matters. You're in command and can decide to land using the other runway because it is safer. The runway seems rather short for a 9kts tailwind landing. Is the airfield controlled? If it is not, you can choose the runway you want to use. I'd choose to land against the wind in this case. When I learnt to fly my instrctor told me that landing lengthes on a POH should be multiplied by 1.3 to get a realistic and safe number. How long would the runway have to be with the 9kts tailwind?
Moreover when you land with a tailwind and there is a wind gust, your airspeed decreases. So you have to have a speed margin because of that and cannot use the short landing speed you usually use, making for a longer landing.
Tailwind and a short runway is a bad idea.
Many pilots have difficulty landing on short runways and will flare longer then they expected themselves above the runway before settling down. I have flown many hours with the PA28 and still do. I have one bad experience with a short runway and a tailwind, that after that I only do this (land with tailwind) if I have a longer runway available. Then I don't care. On very short runways, simply refuse to do it. Divert to another longer runway, use the reciprocal runway or simply wait to see if there is the windspeed as was forecasted (on towered runways you can ask of course for a wind check). Some for takeoff lengths. A tailwind will not be wise on short runways when taking off.
I was flying a C172 last weekend down to about 35 knots with full flaps,
At that speed the AoA is so high, the ASI is not reading the correct speed so you were probably not at 35 Kts!
And a C172 is not a PA28.
In regard to different types - when flying tail wheel aircraft, a little tail wind goes a very long way (pun intended) towards making a difficult situation. The natural directional instability of the aircraft becomes much more pronounced as ground speed increases, and when the rollout starts faster and lasts much longer it can be a very challenging situation even with a very long runway.
