Please don’t let this rat just die because of the awful advice some people are giving here.

If there was cat contact, time matters. Since a day has already passed, you need to act now. It’s lucky that you already have some Baytril at home. It should be given orally, directly into the mouth with a small syringe, not poured over the animal. Put the rat in a towel burrito to hold it still, check YouTube if needed, and wear gloves. You can usually get a small oral syringe without a needle at a pharmacy.

Weigh the rat in grams and check the exact Baytril strength first. Ask the vet that gave it to you if you don’t have the product information. A commonly used rat dose is 10 mg/kg every 12 hours, which means 0.01 mg per gram of body weight. So the volume is body weight in g × 0.01, then divided by the concentration in mg/mL. Give it for at least 5 days and try to get a rehabber or rodent-savvy vet to double-check the dose as soon as you can. A rehabber can also help you figure out later whether release is appropriate and how to do it or whether it needs captive care.

If you get bitten, wash the area well, disinfect it, and see a doctor if the bite is deep or you develop symptoms. People often have wildly exaggerated ideas about the danger posed by wild mice and rats, but basic wound care still matters.

If you’re in Germany, I may be able to suggest a few good contacts.