If it’s not treated, a finger pulp infection can cause a number of complications like spreading of the infection to the bone of the
finger causing osteomyelitis, skin necrosis, septic
arthritis and tenosynovitis (when the infection spreads to the sheath that surrounds the tendons in your finger). Hence my sincere suggestion would be to get it evaluated from a specialist and get the above mentioned treatment.
Hope that this information helps and hope that your sister will get better soon.