Hey let’s not forget the Brat.
And the new Brat. I mean, Baja.
Hooray for subaru!
Emphasis mine.
There’s the rub, we 'murricans think of unit body cars like the Chevy El Camino, Ford Ranchero, Subaru Brat, VW truck and others of those ilk to be pseudo trucks at best. Real pickup trucks have a separate bed to allow for frame flex under heavy loads even if the bed follows the same body controus as the cab. Having separate bodyworks allows for alternate style beds such as the stepdise which was very popular here as well as utility beds like stakesides and tool boxes. I live in Arizona which has the same kind of “pickup culture” as Texas so I think the OP would agree with my definition.
In any event the early model T trucks needed no body modification as they used the same body as the two seat runabout. I’m still looking for better cites but they were offered as a cab and chassis (as is done today so the owner can add a third party bed) and pickup models in the early teens and I found a list of pickup bed part numbers for 1924.
Unfortunately the only online reference I can find is for an authorized Franklin Mint model but the 1913 Model T pickup arguably fits your definition of a pickup box that is integrated into the cab.
http://www.cellinifinegifts.com/fm_1913fordmodeltpickup.htm
Trucks are small? Aluminum? Trucks are made of steel. Things must have changed a lot since I was last in Perth. 
Update:
The helpful mavens of the Model T forum tell me the first complete factory pickup was the 1925 roaster model. Not as early as I thought for a completely factory model with bed but well before '32.
Interesting stuff. Thanks Padeye. Y’see the “first ever was in Australia” thing is one of those little factoids every Aussie schoolkid just knows and the likes of Ford Australia (ironically with a US parent company) have been eager to jump on the bandwagon (the first bandwagown, which was built… ). Now, the cites you provide seem pretty damn solid, and I’ve got a mind to start sending emails off to Ford Australia, motoring magazines, et cetera, and start stirring the pot a bit… 
In Australia trucks are big commercial vehicles, that you move furniture in.
Look at [url = http://www.murphy.com.au/commercial.htm]this car and truck rental site. The ute is at the top, similar to a US pickup truck, followed by a van and a then bunch of trucks (including large vans).
Point is if you say truck in Australia, people dont think of utes, they think of the 1.5 tonne+ vehicles.
Monty - Hmmmm… I was told that, and it made a lot of sense.
I’m surprised that the CHP would be opposed. They don’t seem to be fighting other things that are hard to enforce.
Well considering how different the critters are down under from everywhere else in the world I’m not suprised our taxonomy of trucks doesn’t match up.
FWIW what Nissan and others call a ton truck we call a half ton in the US. Gross weight will be around 6,000lb or around 2,700kg for y’all of the metric pursuasion. Wheels are usually 15" but some Dodges have 16" wheels and with five lugs though some Chevys have six.
A three quarter ton truck has a gross about two thousand pounds more but there’s a lot of variety here. Wheels are normally 16" with eight lugs and axles and frames are heavier. Some three quarter tons trucks will have floating axles but most don’t. A floating axle is where the driving shaft inside the live axle doesn’t have any weight load. If the axle shaft were to break inside the wheel would still be fully supported by the bearing.
A ton truck has bigger gross weight and always has full floating drive axles. They may also come with dual rear wheels from the factory. Aftermarket duelly kits are available even for mini trucks. Wheels are 16" except for a few 16.5" split rims (they require inner tubes) on some Dodges.
Back to the OP I think the tuck culture is uniquely North American. A lot of us in the wes think of trucks as normal personal vehicles. I drive a half ton Dodge and my wife drives a half ton Ford with an extended cab. We both see the practical side of having one regular four door passenger vehicle but nether wants to give up the utility of our trucks.
Dodge USA built a ute-style miniature pickup, based on Omni front-drive mechanicals, in the early eighties: the Rampage .