Should I got to the walk in?!

Should I got to the walk in?

I was holding one of my cats when there was a rather large band outside and as he ran away one of his back claws caught my lip.

So I've got this cut, it goes from about 3 mm up into the skin above the lip to about 3mm into the lip. I don't really know how deep it is I'd guess a couple of mm but now it is all swollen so I can't tell..

It happened about 45 mins ago and is still bleeding a bit. I have been applying pressure.

Just wondering if I should go down to the walk in centre to get it looked at our not...

I would apply ice for twenty minutes, put an antibiotic ointment on it, and watch it for 24 hours. If the swelling gets worse instead of better, you may need systemic antibiotics, lets face it cat claws are not very clean.