their car park is an ice rink
Why was it in that condition? This is key to whether the business could be deemed negligent, and consequently be held liable.
It might be that there was sudden snowfall/freezing rain only a few hours or less before your other half fell over, in which case there may have been little that the business could reasonably have done.
At the other extreme, it might be that the car park has poor drainage with the result that pools of water occur every time it rains and do not drain away, and will inevitably freeze in winter whenever the temperature falls to zero. In that sort of scenario, the business should have been taking active steps to fix the drainage, and in the meantime it should have procedures in place whenever ice formed, such as prompt gritting/salting. In that event, I would think you should most certainly claim: it’s important where businesses are irresponsible and they cause harm as a result, that there is some comeback (even if it’s their insurers that pay), otherwise there is no incentive for them to properly manage the risks on their premises.