USB wireless adaptors at 3.3v ...

I try to find an USB Wireless adaptors working at 5v and 3.3v (When BBB working on battery),
I checking a lot of key without success ...
Someone have an idea ?

Many Wi-Fi modules are designed for a Lithium cell which is 3V to 4.2V.
The first thing the USB part does is reduce the USB 5V to 3.3V
You can boost the battery to 5V with something like

Looking at BeagleBone Black rev C schematic and TPS65217C data sheet

If the battery is connected to the Rechargeable Battery access pins (TP5/6/7/8 TESTPT1),
The battery feeds the USB HOST VBUS ‘5V’ through SYS_5V, the 3V to 5V booster would have to be an inline adapter plugged into the USB HOST connector, this is slightly non trivial.

In theory, a booster input could be connected to SYS_5V on P9.7 (ground to P9.1) and booster output to an USB hub power input with Wi-Fi plugged into hub, however many hubs let the 5V hub power flow back into the host.

If a battery pack is just connected to the DC in Jack, the voltage should be 4.3V to 5.8V.
The battery pack could be connected to a 5V booster input and the DC jack to booster output.

When the BBB work on battery and come back to 5v the USB port doesn't work !?

Sorry to misunderstand your question, if French is your first language, could you post in French and English.

Is “BBB” a BeagleBoneBlack (non Wi-Fi)? What is “BBB” abbreviation for?

How is the battery connected?

Does the BBB recharge the battery?

Does the USB Wi-Fi work when the BBB is powered through the USB PC MINI connector?

If not, it might be using too much current.

I do not know any USB devices that work below 4.4V. I would think, USB devices need > 4V for its own 3.3V supply.

The BeagleBone BATTERY Cape, supplies 5V to the DC in jack through expansion connector P9.5 and P9.6. The BBB should work like connected to 5V power supply to the 5V DC in Jack.

You will need to use alkaline batteries (cells) or rechargeable NiMH to get enough current; (the cape has the batteries connected in parallel, AFAIK you should not do this; the Tesla battery pack has each cell connected through PTC resistor/fuse). For the BeagleBone and USB Wi-Fi dongle to use 500mA, the batteries would have to supply 1A.
Perhaps, the Board ID Memory says it cannot supply enough power for USB host.

BBB = Beaglebone black
BBBW = Beaglebone black wireless

Battery connected to TP5,6,7,8 (LiPo 3.7V)

USB Wifi work with 5VDC (Jack)

The physical address of MUSB_DEVCTL for MUSB1 is 0x47401C60, I used memtool to play with it ...

And now, the USB Wifi key works on battery ...