I have an existing shower stall with metal walls and a precast
concrete base.  The concrete base is stained and I would like to
refresh it.  My question is:

Is it possible to successfully paint the concrete base in a way that
will ensure that it won't peel or chip over time? Also, the surface
mustn't be too slippery.  If so, please list the steps I must take and
the materials I must use.  Please back up your answer with references.

Start by removing all the soap and minerals that are on it. Some
non-slip floor paint might work.

I know just what you need! Epoxy based paint! Just ask for it and a
paint store, the good kind comes in 2 cans that you have to mix
together. IF IT'S NOT 2 PART YOU DON'T WANT IT! You can also add sand
to it to give it texture. I've used a fair amount of this stuff to
paint garage floors. It's tough stuff and won't go anywhere once it's
dried. If you want to go all out you could go for the marine epoxy,
but the regular epoxy from the paint store is more than good enough
for your needs.

Basically for install you'll want to sweep up, then etch the concrete.
You can get some muriatic acid at any hardware store for this. Just
pour it on and scrub it around with a good stiff brush like a push
broom, or whatever you can find this will make the floor a bit rough
so the paint sticks better.

Then clean it with a degreaser like Purple Power and rinse it out with
plenty of water, you don't want any leftover residue.

Then let the floor dry. At least 24 hours of not a day or two you
don't want any water trapped underneath the paint.

Then mix up you're two 1 gallon pails, Add sand or whatever you want
to use for texture to is and STIR WELL. The amount of sand will vary
depending on you. If you find it's not enough once you start just
throw some more on the floor and roll over it with you roller until
you're getting the desired results.

Paint around the sides with a brush and then roll the middle. Be sure
and not to use a good roller, or brush and don't wear anything you
want to keep, including shoes, cause once this stuff is on it's on,
theres no getting it off with mineral spirits. You may want to wear a
long sleeve shirt and gloves as well cause it takes a while to get it
off your skin also.

Let it dry a good 24 hours before walking on it. Considering this is a
shower and people will be walking on it barefoot, or with flip flops
only it'll probably out live the shower, as this is the stuff they use
on the bottoms of boats and garage floors.

I don't have any references other than myself. I spent many of my
undergrad years as professional painter (I was on the 8 year program)
mostly residential repaints in the high end neighborhoods here. If you
have any questions feel free to ask.