We have two female Shih Tzu. The older resident (Millie) is now 15 yrs., and we've had her since 8 weeks. During all this time she has been the only dog, and has been the model companion - no trouble at all. She has always been laid back; indifferent to other dogs, she has always been a "people" dog.
We now also have a 9 week old puppy (Bella). Typical puppy, everything and everyone is a game. Unfortunately Millie doesn't want anything to do with this, and when Bella wants to play, Millie goes upstairs and hides.
We feel that Millie's quiet old age has been spoilt by us , because she's so "soft" she won't exert any authority over the puppy, and the puppy just won't leave her alone. This results in a barking confrontation, with Millie running out of steam and then hurrying off to hide.
Any suggestions so that Millie can get some of her quiet life back, and Bella can just "back off" a bit?

Your older dog doesn't want to have to tell your puppy off, it's up to you to control the pup and to limit access to your older dog. Set up stair gates at doorway entrances so that your older dog can still be with you on the other side of the gate, but protected from the lively puppy. Supervise the puppy with your older dog and make sure she isn't constantly harassed by the pup, step in and intervene before she feels the need to go and hide.

I agree with Jodi, puppies can be full on, so it is up to you to let the puppy when to cool down, the other thing you can do is redirect the puppy away from your older dog, and get the puppy to focus on you.

As has been said, to make this work, it's YOU who has to correct the puppy when she gets too much for your older dog.   Adults won't normally attack puppies, but if they don't see you doing what's needed then she might, or as is happening, run away which is very sad at this point in her life.   It might have been kinder to let her live out her days before bringing a new puppy in.    Only you can see how upset your oldie is with all this and if you feel it's too much, perhaps take the puppy back until you no longer have your oldie?

For now, you do need to make a quiet place for your oldie to retreat to when needing a nap, and use a crate for the puppy when you can't supervise, for now again.

I found a crate with an extended pen invaluable - for the puppy not the older dog. It took over the room quite a bit, but was worth it. They get used to having each other there, as well as you, and learn to relax and play separately - not forgetting that young pups also need a lot of sleep. I always supervised any play/interactions to make sure they were positive. I use distraction, such as a tugging toy to redirect excessive play by the youngsters.
When I got my little terrier mix with my 8yr gentle Elkhound I also used an Adaptil plug in, I don't know whether it worked but they were great mates and she wasn't particularly bothered about playing with other dogs, apart from him. She would stand in the middle of the room and just playbow at him and then he'd do the zoomed and so on :D
I just feel it's important that the existing dog isn't disturbed too much and teach the pupils to fit in - obviously not neglecting pup but they do need a lot of sleep
I hope things work out for you ;if not, returning the puppy sooner rather than later may be best.

The other thing that's useful, is a puppy house line.  It's a very lightweight 2.5m lead your puppy can trail around the house and you can grab the lead and gently distract or remove puppy from older dog.  Look on Amazon for puppy house line...

When I'm integrating a pup into my group of dogs, we do lots of 'hanging out' on the sofa in the evening, with me using the house line to prevent pup from harassing old unappreciative dogs and providing chews or stuff for the pup to do instead.  It does take a few weeks, though - but eventually the pup will learn.  No need to 'correct' the pup at all, just use prevention - crate, house line etc.