where can I locate and purchase a portable air conditioning unit which can easily fit on a car dashboard or office desk?

An air conditioner has to get rid of heat.

In a building, you can get (used to be able to get) a unit that fits on a window sill, is sealed around the surrounding outside of the unit and then it cools inside and exhausts heat outside.

I do not know of any safe automotive unit unless it is connected to a refrigerant compressor in the engine compartment.

The refrigerant rules are very stiff, so ALL automotive units I have seen are built in in the factory.

For window units look at something like this:
https://www.rona.ca/en/horizontal-air-conditioner-12-000-btu-15625013?viewStore=55070&gclid=CjwKCAjwmufZBRBJEiwAPJ3LploH13IPqjcfHiOScNONa5gOVQV5puY5__mUG5jGCbHzqUr-VsCJ5xoC6FwQAvD_BwE

There are new designs of room air conditioners.

https://www.wayfair.ca/home-improvement/sb0/air-conditioners-c420527.html?refid=GX229395339783.Window+airconditioners%7Ee&position=1t1&network=g&pcrid=229395339783&device=c&targetid=kwd-315406269&gclid=CjwKCAjwmufZBRBJEiwAPJ3Lplu7VNDM0AR5OudOWoEq3WfNIePzEmD9TfV7u0UNGJJSdY7K_8200xoCpHEQAvD_BwE

You need something that does not require a condenser.  This unit will work by placing ice in its water container, but of course it has limitations...

https://www.amazon.com/gp/product/B07B4BKHGD/ref=s9_acsd_zwish_hd_bw_b50Ws_c_x_w?pf_rd_m=ATVPDKIKX0DER&pf_rd_s=merchandised-search-8&pf_rd_r=8JHN7Y5BFW044M4TVQ2B&pf_rd_t=101&pf_rd_p=38a7dfee-97a1-512e-b681-f15c5f252528&pf_rd_i=1193678

In a hot car (+35 degrees C in the sun) the ice would not last very long.

You really need a built-in unit running on the car engine for automotive use.

If you saw a device that was described as a dashboard-mounted air conditioner, then the manufacturer is undoubtedly exercising some latitude with the term "air conditioner" and it will likely be nothing more than a particulate filter and fan to "condition" the air in the car.  That's what the car's replaceable pollen filter does in combination with your standard blower fan.

I just came back from washing both my cars. HOT outside and A/C was running. You could feel the heat from the front of both cars which was the A/C compressor and stuff dissipating HEAT.  

No such practical item as  portable car air conditioner. Nothing like that exists.

Hello and Good Evening Everyone,

           Thank you very much for your thorough feedback in addition to your resourceful links.   I did locate the small or portable cooling units that came to my mind.  They are located at the following links:  https://l.facebook.com/l.php?u=https%3A%2F%2Fwww.target.com%2Fp%2Fas-seen-on-tv-arctic-air-space-cooler-white%2F-%2FA-53216700%3Fref%3Dtgt_adv_XS000000%26AFID%3Dgoogle_pla_df%26CPNG%3DPLA_Kitchen%2520Shopping%26adgroup%3DSC_Kitchen%26LID%3D700000001170770pgs%26network%3Dg%26device%3Dc%26location%3D9060073%26gclid%3DEAIaIQobChMIwIjSwuGB3AIVF3ZeCh0CAw1nEAQYAyABEgI_GfD_BwE%26gclsrc%3Daw.ds&h=AT06fwLT_sHGXysIzdW3EGxii1n3CSrsEoSZ0uMie_aof9lPUBI-o2WA88W7cmT7uDAJvAvk4zewQpgXZ6RbrJzRvO0F-PdRH716Iq5nIz7lo1ioIBx-8zQ3gefoM0StRs4D  and https://www.amazon.com/dp/B07B4BKHGD/ref=asc_df_B07B4BKHGD5523697?tag=shopz0d-20&ascsubtag=shopzilla_mp_1049-20&15306394449413453910010080301008005&creative=395261&creativeASIN=B07B4BKHGD&linkCode=asn  Seeing that the reviews are most mixed, I decided to not carry out any further follow up on them.

George

These links are basically a rehash of Fred's given link.  For a cooling unit to have long-term and dependable performance,  it must rely upon the science of thermodynamics as outlined by David.  As mentioned by John, the units which I had in mind would not hold up too good under extreme heating conditions within a vehicle during summer months.

The arctic air uses evaporative cooling which actually raises the humidity, not something you'd want in a humid environment.. in the desert maybe but not near the coastline or great lakes