Upgrading cassette from 11-28T to 11-32T?

Frame size doesn't really matter.

11-32 does make a substantial difference; remember its the ratio of the front gear to the rear gear that matters. Your lowest gear will be 12.5% lower. Also, you'll get a wider (and likely more useful) spacing in gearing between changes.

You will need a new chain as well, but it should be fine to install. Putting a bigger cassette on is a good starting point.

Of course, you have to shift appropriately -- if you're running 50/11 all the time, you're going to be tired regardless of what the biggest cog is. A 11 small cog is not very useful all the time.

Finally, "extremely steep" has a different meaning to everyone and their abilities (and the type of biking they're doing). For example, a touring bike in the mountains might be using a chainring in the 20's of teeth and some huge cassette to get the gearing needed to comfortably ride up the mountains. Most racing bikes aren't really built with that kind of riding in mind remotely.

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Also, if you're using a 47 cm frame and it has 700c wheels, you can also get a reduction in gearing (and height) by switching to 650b wheels (easy if you have disc brakes; bit more work if you have rim brakes, but might be possible). My personal view is people below 5'5" ish shouldn't be riding 700c wheels.

Let us assume that you weigh 70 kg and the bicycle weighs 15 kg.

70 kg at the pedals for 170mm cranks is 116.74 Nm.

34T chainring and 28T rear sprocket translates 116.74 Nm into 96.14 Nm.

The rear wheel has about 339mm diameter so there is 283.6 Newtons or 28.9 kilograms of force when cranks are horizontal. To get average propulsion force over one crank revolution, multiply by 2/pi, so there is 18.4 kilograms of force.

This allows climbing hills up to 18.4/85 = 0.216 grade, i.e. 21.6% grade. Such hills are extraordinarily rare. Where I live, 10% hills are common but over 20% hills are not.

If the minimum speed at which bicycle stays stable is 10 km/h or 2.78 m/s, then climbing a 21.6% hill at this speed would require 501 watts or about two thirds of a horsepower. A bicyclist producing 501 watts would tire out very quickly.

I suspect the limitation here is not lack of gears but rather lack of power. Going up a steep hill is always difficult. No amount of low gearing is going to change that.

Yes it will make a difference, it's a difference of cadence. Now, you won't be any more powerful and you will have the same minimum speed before you lose your balance, but when you ride slowly you can use a higher cadence. For example at 10 kph you could be riding at a cadence of 74 instead of 65; this is a big advantage in the real world and your legs will feel looser for longer as the hill gets tougher.

In this context, 32 is noticeably bigger than 28. Imagine riding up the same hill without using 28 and using the next smallest gear instead - you'd be cheating yourself out of an 'easier' gear. By not fitting the 32 you're doing just that.

If you try the 32 and don't like it, nothing is lost, you can easily put the 28 back on.