The problem is solved! Thanks to user1686!
This issue was really caused by the Wi-Fi driver "brcmsmac" (Broadcom Wi-Fi driver).
To resolve this issue I decided to install another proprietary driver for my wireless card because I require Wi-Fi on a daily basis.
At first I plugged my Android to use it as USB-tethering to avoid crashes.
Then I downloaded another module for this wireless card:
# pacman -Syu broadcom-wl
The problem module brcmsmac was loading as part of initramfs on boot. I checked this by using the command:
~$ mkinitcpio -M
I blacklisted this module (and "b43" module that is also in conflict with "wl" module).
In order to blacklist these modules I created a .conf file inside /etc/modprobe.d/
directory:
# nano /etc/modprobe.d/blacklist.conf
And then I appended these lines to the file:
blacklist brcmsmac
blacklist b43
And then, guided by Arch Wiki, I added the path to this .conf file to FILES array in mkinitcpio.conf:
# nano /etc/mkinitcpio.conf
FILES=(/etc/modprobe.d/blacklist.conf)
Then I regenerated the initramfs and rebooted the system:
# mkinitcpio -p linux
# reboot
Once the system was loaded I tried to connect to my Wi-Fi network (I use NetworkManager) and it worked!
But still I was needed to test my connection, so I tried to download some huge file from the Internet and nothing bad happened, no crashes and the file was completely downloaded.
I don't know if I did everything right but now my laptop is working stable. I watch videos online, download files, upgrade my system using pacman, no kernel panics so far!