I have a S&W Model 38 with a Nickel Finish. It has one small chip on the finish. It there an easy way to repair the chip or something that can be done to prevent the chip from becoming any larger?

Sorry, once chipped, scratched, or otherwise damaged, there's nothing you can do.

There's no way to "fix" a chip, and all you can do to is to keep a light coat of lube or wax on it to prevent moisture from infiltrating under the nickel and rusting the steel.
As long as you don't hit the area it shouldn't flake off or spread.

For wax use Johnson's Paste Wax or Renaissance Hard Carnuba.... NO CAR WAX.

I have had some success with using a polishing wheel on a Dremel with some Flitz to brighten up the chipped area. If you polish the steel well enough it will get bright and smooth enough that it is almost indistinguishable from the Nickel. You do have to be careful do not burn or chip the guns finish any more than it is now. I just did this on my inherited M36 (1951) Chief's Special with excellent results. The Nickeled screws were quite bunged up, and chipped. After trying hopelessly to buy new Nickel screws for months, I gave up! I used a small peening hammer to reshape and repair the screw heads, a set of jewelers files and Arkansas stones to re-cut the slot and make them round again. After I was satisfied with that, I highly polished them with Flitz and a Dremel and they look just like the Nickel finish. Just go slow and have a lot of patience, and you be OK.

Regards,
Chief38

Spot plating does exist but whether it is adviseable for the described condition I don't have that kind of experience with it. Texas Plater Supply sells a kit that does electroplating with a brush, a chemical paste and a 6 volt DC power source which can be flashlite batteries. I have used their product at least 25 years, as recentlly as maybe two years ago. I have plated several complete guns and have touched up worn areas on many others.

If a chip has flaked off a gun I think the procedure would be to spot buff the area to blend the existing plating with the steel of the gun, degrease/clean the area and do the brush job. Without good prep, at best it would look like a nickel patch.

If you are comfortable with the work and/or willing to build up some experience on scrap it could work for you.