The BMW i3 isn't built like any other electric car. It's made of different stuff, and its layout is different. Everything about it has been considered and re-thought. Despite the fact it comes from an established car company, its whole approach and execution is amazingly fresh. And so driving it is a mind-cleansing experience too.

It makes you feel better. It really does. It's one of those cars that changes the way you drive. Among BMWs, the i3 isn't at all expensive. But actually it's a bargain because it gives you a whole new you.

While the Nissan Leaf or Tesla Model S or Vauxhall Ampera all look like modern conventional cars, the i3 is designed to reflect what's underneath. There's no attempt to hide the narrow, low-drag tyres, or the advanced aero. No attempt to hide the fact that it's high because the battery is under the floor, or that it's snub-nosed because the motor and gearbox are in the back.

That's because it's a car for cities and suburbs, where you really can gain by being in a car that's short, and has a high eyepoint and a glassy cabin and vertical sides. You can see what's up around you, and be a bit cheeky about squeezing into gaps. And when there are no gaps, you can at least enjoy the view. I was in Holland, and drove the i3 down some scenic, if tediously flat routes (or enchantingly big-skied, if that's how you see it), and enjoyed the view. Then into the centre of Amsterdam, and was glad I could see the trams coming and weave through the bedlam of bikes pinging at me from every conceivable direction.

The interior, too, gently smoothes the rough edges off your mental state. It's distinctly non-automotive, but in a good way. The floor's flat, and the dash doesn't have the usual consoles and divisions between the people. The materials and the shapes are like some slightly groovy domestic room set.

None of that would be worth dwelling on if it didn't perform properly. But it's got your back there too. It's eager and silent, and taut and biddable. Electric propulsion is conceptually simple, but in its details it's extremely hard to get right. And in mass-market terms it's in its infancy. The depth of BMW's engineering achievement mustn't be under-estimated.

In most normal suburban driving the i3 is somewhere between quick and actually fast. Normally it eases its way from rest with impeccable smoothness, but floor the thing and it departs as if high-voltage electrodes have been applied to its derrière. Which of course they have. OK, listen hard and there's a slight whining noise, but overall, even compared with other electric cars, it's miraculously silent at town speed.

The silence isn't just from the motor. Other electric cars have humming high-voltage electronics and whirring fans. This one packages all that stuff in places where you don't seem to hear it – even the air-con and heating unit is ahead of the bulkhead, whereas practically every other car parks it right there in the dashboard. And the narrow tyres don't kick up much racket either. Wind noise is the biggest thing on a motorway, but only because everything else is so subdued.

That suave push from the motor just keeps flowing. The transmission is a one-speed reduction gear, so it's smoother than a manual, smoother than an auto, smoother than a twin-clutch. Totally smooth, end of.

Smoothly, at somewhere above 70-mph, the motor's acceleration begins to taper away compared with your expectations. This is a 170bhp car sub 60mph, but at bigger speed it isn't. That's precisely because it has just the one gear, and it's now revving beyond its power peak. Anyway, to prevent energy-sucking high speed running, it's limited to 93mph. But it's not a deal-breaker: you can move into the outside lane without it betraying you.

The acceleration is better than most e-cars because it's so light. BMW has been researching for a decade how to reduce the cost of carbonfibre. With the i3 it has got to the point of using an entire carbonfibre upper body cell, mounted on an aluminium chassis. This cuts the weight considerably. And lighter weight means it needs less battery for a competitive 90-mile range. Since batteries are expensive, the saving on battery pays for the carbonfibre. And a smaller battery is lighter too. Brilliant. We said this was built entirely differently from everyone else's electric cars. It weighs 1200kg, which is 350kg less than the Nissan Leaf, which gets by on just 109bhp. That's why the 170bhp BMW goes so well.

So that's acceleration. Guess I should go on about the brakes now, but to be honest I never noticed them. I drove for miles on end without ever touching them, even in traffic that was changing speed, even when I came to traffic lights and junctions and roundabouts from dual-carriageway speeds. Instead, the slowing comes from simply lifting off the accelerator.

Back away and the slowing-down smoothly gets more emphatic. The brake lights come on too, by the way, so you're not a traffic hazard. The last few yards are wonderfully supple, the car just arriving at a stop without any kind of shunt. It avoids that little full-stop jerk you get with a normal car's friction brakes.

Drive like this, keeping the brake pedal as little more than backstop for brutal pull-ups, and you'll be harvesting energy nicely, feeding it back to the battery.

And it corners like a BMW. Sort of. OK, a short, wide BMW with narrow tyres. Pile into a bend and it'll understeer. Jam the accelerator hard in a tight bend and the front end goes light and it'll understeer. But be smooth, or give a slight lift to dig the front tyres in, and it's neutral, the driven rear wheels finding plenty of traction. The chassis gives you good feel for what's up, the steering less so. But the steering is direct and the wheelbase short, so the i3 is always agile.

Because you sit high, there's some lateral rocking on undulating roads, but nothing to upset the applecart. Otherwise the ride is decently controlled, if fairly taut. Because the body feels so strong and rigid, you've got confidence.

Confidence comes too from the infomatics. The satnav is the first on any car that will sometimes tell you NOT to drive. Just select the 'multimodal route' option and it'll automatically go online and figure out if driving might not be the fastest way, or if your destination is beyond your range. If so, it'll find you a charge point (a vacant one) near a bus or rail stop, and tell you the timetable, and the walking directions at the other end. It'll then automatically upload these to your personal BMW account, which sends them to your phone. So you park the car, then walk off following the directions on the phone.

That makes a flat battery less of an issue, so 'range anxiety' withers. Also, in my drive, the i3's instruments turned out to be extremely accurate in predicting remaining range. And amazingly, for the first 55 miles I used almost the same number of kWh as I should have according to the boring drones from the EC cycle test. Next day I drove impatiently (late for a plane), and going like that for a whole battery's worth would have suppressed the range by about 25 miles.

Then there's the option of a little range extender under the boot floor, a two-cylinder motorbike engine humming away with a generator attached to maintain the battery charge. With a 9.0 litre petrol tank that gives another 60 miles. But don't think of it as an everyday electric car. After all, 60 miles off a nine-litre tank is an unimpressive 30mpg.

For your occasional long journeys, BMW will swap your i3 for something else, a 320d maybe, for a week or however long you need. Or a Z4 for a sunny weekend. An X5 for a skiing trip. Whatever. It's all there in the lease package options.

So yes, it's an electric car, and range is a limitation. But it needn't be scary. The car is very precise at letting you know how much you have left, and offers you options (trains, range extender, another car) as get-out-of-jail-free cards. Of course electric cars won't do unexpected 500 mile journeys, just like two-seaters won't unexpectedly transport a family. You know that when you buy them and if that limitation is too much for you then shop elsewhere.

If the i3's range fits your life, here's what you get. A car, a gadget, a suite of furniture, a greener option, a talking point. And a slower heart-rate.