Severed fingers can last for a period of 12 hours on ice.
If the severed finger is to be attached again surgically to the person then the finger must be attached within 12 hours and preferably before 12 hours.
After 12 hours even when on ice the finger can get bad enough because the tissue degrades and then putting the finger back on the person would not be possible.
So it's important to get the finger reattached to the persons hand within a few hours to no longer than 6 hours.
When the finger is severed the tissues will survive for about six hours without cooling and If the part is cooled, tissue survival time of the severed finger is 12 hours.
After 12 hours the finger tissue dies off.