Try it and see. It might work. Your best chance is to get the antennas high up (>2m, but roof level if you can). Avoid any vegetation in between.
If you have a directional antenna to link to the cottage you may need an omnidirectional antenna to provide all round coverage at the remote end, although lecage from the directional antenna may be adequate.
It sounds like you will need a device at the remote end that can act as a Wi-Fi repeater. Some routers may not do that by default. Check the manual for repeater setup instructions. If that isn't supported you may be able to change the firmware to run openWRT
or the like which provide lots of options and run on lots of common hardware.