Yes, a doctor needs to prescribe you some antibiotics right away.
Wounds from cat bites are some of the most infected wounds you can get -
mainly because cat teeth bury bacteria deep down in tissues not prepared
for this attack -  because cat teeth are so thin and needle-shaped.
So see a doc ASAP!

Niels Peter

many year ago when my daughter was about 6 she ran in the house stepped on
our old cat and he bit her on the ankle........ I cleaned it (had 4 fang
marks) then thought no more of it, when she went to school the following day
it started to swell up and go very red and hard, I took her to our GP and
they put her on antibiotics. It did take a few days to settle down.

My cat bit me on the hand once while at the vet's.   The vet bill was only
$20, but, it cost me about $120 for antibiotics and a tetnus shot when my
hand swelled up *alot* the next day.  Linda

Bitting is bad behaviour and should never be tolerated, Bill. Not for
one second. Why do you allow it? By not acting, you re-enforce her bad
behaviour.
 I have found the answer, even though none of these wishy-washies
here wants to admit it. My can NEVER bites me now. He used to, but not
anymore. He spends his time now bidding for my attention and
affection. I'm telling you the truth. I can see when I scold him that
he would like to bite me sometimes, but he won't dare. I sometimes
test him by petting him when I know he doesn't want me to. When he's
in a bad mood, and he just wants me to leave him alone. I purposely
pet him just to challange him, but still he won't even try. I only do
this every once in a while, just to make sure that he is still
conforming. It's been over a year now, and I've not been bit once. But
the wife has, and she's the one who feeds him! (she always shakes her
head on that).
That's just the way it is. Strong messures must be taken to defeat
strong wills.

IBen

I didn't read this bit at first, but: if your finger is swollen, get to
a doctor NOW as in NOW!

Niels Peter

Have you had a tetanus shot?  In any case, I think it would be a good idea
to check with your doctor since the finger is swollen.

MaryL

I got bit by a cat years ago when I grabbed it from a dog in order to put in
back inside. I washed it, applied creme etc. Within 24 hours I was in the
local hospital with blood poisioning. I was admitted and spent a week in
hospital and had surgery to remove the nails from my thumb and forefinger. I
almost lost the finger due to infection. Go to a Doctor now!!!!! The only
bite worse than a cats is a humans. DO NOT WAIT ANOTHER MINUTE.

Cat bites can be nasty.  Those teeth are perfect inoculation needles.
I assume you scrubbed the wound well with soap and water.  If you
haven't had a tetanus shot in a long time you want to get a booster.
If it is getting infected you really need to have it looked at.  Some MDs
put animal bites on prophylactic doses of antibiotics regardless, but
there is a difference of opinion on the need for that.  The last time I
was bitten by a dog the ER doc prescribed antibiotics.  This was work
related and dog bites are reportable so I took myself to the medical
center immediately after.  That @#$% dog was a dedicated biter.  It ran
up without making a sound and got me on the hindquarter, drawing blood
and ruining a pair of pants.  I healed without incident.