As the title says, according to my Smiths volt gauge, the voltage steadily and slowly climbs as I drive.

When I start the car, it reads approx 13.5 volts, and the speakers aren't whining at all. After 15 minutes, it will have climbed to 14.2, and if you listen carefully, the speakers have a bit of a high-pitched whine. After 30+ minutes, the voltage is closer to 15, and the motor noise is drowned out by the speaker-squeal.

The speakers are not connected to anything. There is no head unit installed in the car.
If I turn off the car for even 1 second, the voltage resets to 13.5 and I've got 30 squeal-free minutes.

What would cause this condition?

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1998 Rover Mk VII
2001 Honda CBR 929RR (street)
2014 BMW S1000RR (track)

If it's getting to 15.0V or more your voltage regulator probably needs replacing (if it's got an alternator).
If it's running a generator, the voltage regulator needs cleaning & adjustment.

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DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.

It's a 98 Rover SPI, so most certainly running an alternator.

Excuse my ignorance, but where wold I find said voltage regulator? My first instinct was to search the parts catalogs (MiniSpares and the like), and drill down into the blueprints to see where this thing lives, and how to replace it. Unfortunately I didn't come up with any hits. I can only assume it is part of the alternator itself?

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1998 Rover Mk VII
2001 Honda CBR 929RR (street)
2014 BMW S1000RR (track)

Pretty sure it lives behind the speedo

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SPO41 Mk2 Cooper S very original
1993 R32 Nissan Skyline GTR

Isn't that the Voltage Stabilizer? And I thought that was more of a function for the gauges themselves, not the electrical system of the entire vehicle...

Hopefully the doc will chime back in and clarify.

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1998 Rover Mk VII
2001 Honda CBR 929RR (street)
2014 BMW S1000RR (track)

Nope that is the voltage stabilizer (for Smiths gauges).
The voltage regulator is inside the end cover of the alternator. They do sometimes go belly up.

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DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.

When you say "Slow Climbing" do you mean it takes a long time to climb to the around 14Volt mark or slowly climbs too high, like over 15 Volts. If the first one check the Alternator belt and if it is an Air conditioned model check the lower belt too as the Alternator is driven off the AC compressor pulley and the lower belt is often ignored as it is difficult to adjust for some. I have in stock all Mini Voltage regulators if you need one. Except the external one for the K and GT.

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DID I MENTION THAT I AM THE BEST SUPPLIER FOR MINI ELECTRICAL PARTS AND REPAIRS?
"Let us remove your "SHORTS"

Slow climbing as in, when I start the car up it is where I'd expect it to be (13.5 - 14 range). As I drive, it slowly gets higher and higher - over a long period of time, like 30 minutes. I don't think I've ever seen it over 15, but certainly darn close.

I've never opened an alternator before. Is this a tough job? Is there anything else worth looking at / cleaning / repairing while I've got it out?

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1998 Rover Mk VII
2001 Honda CBR 929RR (street)
2014 BMW S1000RR (track)

The alternator fitted from 1997 onwards is from a land rover- you cant dismantle and rebuild it like a japanese one (mainly because parts arent available and also cause its soldered together. From memory attempting to pull it apart found the regulator was thr only replacable part). Mine died at about 90,000 kms.....commutator had worn down and was squealing as you say.

I still have the old alternator here: you can have the regulator from it to fault find if you like?

Hang on you say the speakers arent connected to anything?
You know the fuel pump can make a high pitch buzzing /squealing sound when its on the way out too.....which funnily enough sounds like its coming from the speakers on the parcel shelf.

the A115i alternator is quite easy to repair, all parts are available (at least over here). yes, you have to de- and resolder some things if you go beyond replacing the regulator. and having some special tools will help a lot.
one may argue if it is cost effective, but over here a new one is very expensive and professional rebuilds may not be better than the worn one

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Rover 214 atm ....

I had to get the parts to rebuild mine from overseas......it was only going to be a small amount more for a whole new alternator so thats what i did.

When the brushes and regulator need replacing it is usually time to do the bearings as well.

It was common to most early Lucas alternators that you had to desolder the slip rings to get the rear bearing off.
But at least you could replace the brushes without pulling it apart.

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DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.

Use a proper digital multimeter to monitor the voltage incase the gauge is reading incorrect