Yes, cats can very definitely get ingrown nails. Pretty much all you can do is to (a) remove that part of the claw which has penetrated the paw pad and (b) make sure that there is no infection in the pad and that no infection takes hold and (c) prevent it happening again by regular claw trimming or by making sure the claws are worn down through use.
Ingrown claws are particularly likely to happen in older cats for the obvious reason that they are far less active therefore their claws are much less likely to be worn down. In normal use the claws of a cat are worn down and therefore there is little or no likelihood of the claw growing long enough to turn into the paw pad.
However, I am sure that it is quite common for older cats to develop an ingrown toenail which is long enough to pierce the paw pad. Sometimes it can pierce quite deeply into the pad causing considerable discomfort and pain. It is remarkable that a cat will not show discomfort in his/her paw until quite late in the day.
Today, a lady visitor to this website, Signe, had this problem to deal with. Her cat is 17 years of age. All of his claws are too long and one had grown into the paw pad causing much pain, she said.
I, myself, had to deal with this some years ago. My female cat who is now deceased developed an ingrown toenail. This was my fault because quite obviously prevention of ingrown toenails is far better than fixing the problem. The lesson is that in an older cat the owner should regularly, at least once a week or perhaps more frequently, check their cat’s toenails and trim them.
In my case, I decided to trim the offending toenail. I asked a close friend to hold my cat in a towel and I used cat nail clippers to cut off the end of the claw which had pierced the paw pad. This caused a lot of discomfort and my cat objected to it but the job was done. The paw pad healed on its own without treatment. The problem was solved. Note: this mini-operation needs to be done with precision, obviously. You can’t cut off too much of the claw otherwise you’ll cut into the living part of the claw and neither can you cut the paw pad. If you are honestly unable to be precise enough for whatever reason, don’t attempt it. The fact that the end of the claw is in the paw pad makes this procedure tricky.
Perhaps the best or safest advice is probably to consider taking your cat to a vet to deal with this because it is possible that the paw pad may have become infected.
As for painkillers, as soon as I had removed the offending part of the claw, my cat was no longer in discomfort so pain medication was not required and the wound healed quickly without treatment. If you think that pain medication is required I would avoid giving your cat a drug called Metacam as it can cause kidney damage. In an older cat who may be developing kidney failure in any case, to give this drug to him/may prove fatal.
As to any other painkillers I would always seek veterinary advice because as we all know giving painkillers to cat is a tricky procedure and great care must be taken.
If visitors can add to this please do so in a comment. Thanks.