What's the output of:
sudo libinput-list-devices

And you wouldn't happen to have the xf86 synaptics or evdev drivers installed as well?

``sudo libinput-list-devices`` gives:

Device:           Power Button
Kernel:           /dev/input/event6
Group:            1
Seat:             seat0, default
Capabilities:     keyboard 
Tap-to-click:     n/a
Tap-and-drag:     n/a
Tap drag lock:    n/a
Left-handed:      n/a
Nat.scrolling:    n/a
Middle emulation: n/a
Calibration:      n/a
Scroll methods:   none
Click methods:    none
Disable-w-typing: n/a
Accel profiles:   n/a

Device:           Video Bus
Kernel:           /dev/input/event16
Group:            2
Seat:             seat0, default
Capabilities:     keyboard 
Tap-to-click:     n/a
Tap-and-drag:     n/a
Tap drag lock:    n/a
Left-handed:      n/a
Nat.scrolling:    n/a
Middle emulation: n/a
Calibration:      n/a
Scroll methods:   none
Click methods:    none
Disable-w-typing: n/a
Accel profiles:   n/a

Device:           Sleep Button
Kernel:           /dev/input/event5
Group:            3
Seat:             seat0, default
Capabilities:     keyboard 
Tap-to-click:     n/a
Tap-and-drag:     n/a
Tap drag lock:    n/a
Left-handed:      n/a
Nat.scrolling:    n/a
Middle emulation: n/a
Calibration:      n/a
Scroll methods:   none
Click methods:    none
Disable-w-typing: n/a
Accel profiles:   n/a

Device:           Integrated Camera
Kernel:           /dev/input/event18
Group:            4
Seat:             seat0, default
Capabilities:     keyboard 
Tap-to-click:     n/a
Tap-and-drag:     n/a
Tap drag lock:    n/a
Left-handed:      n/a
Nat.scrolling:    n/a
Middle emulation: n/a
Calibration:      n/a
Scroll methods:   none
Click methods:    none
Disable-w-typing: n/a
Accel profiles:   n/a

Device:           Microsoft Microsoft® Nano Transceiver v1.0
Kernel:           /dev/input/event1
Group:            5
Seat:             seat0, default
Capabilities:     keyboard 
Tap-to-click:     n/a
Tap-and-drag:     n/a
Tap drag lock:    n/a
Left-handed:      n/a
Nat.scrolling:    n/a
Middle emulation: n/a
Calibration:      n/a
Scroll methods:   none
Click methods:    none
Disable-w-typing: n/a
Accel profiles:   n/a

Device:           Microsoft Microsoft® Nano Transceiver v1.0
Kernel:           /dev/input/event2
Group:            5
Seat:             seat0, default
Capabilities:     keyboard pointer 
Tap-to-click:     n/a
Tap-and-drag:     n/a
Tap drag lock:    n/a
Left-handed:      disabled
Nat.scrolling:    disabled
Middle emulation: disabled
Calibration:      n/a
Scroll methods:   button
Click methods:    none
Disable-w-typing: n/a
Accel profiles:   flat*adaptive

Device:           Microsoft Microsoft® Nano Transceiver v1.0
Kernel:           /dev/input/event3
Group:            5
Seat:             seat0, default
Capabilities:     keyboard pointer 
Tap-to-click:     n/a
Tap-and-drag:     n/a
Tap drag lock:    n/a
Left-handed:      n/a
Nat.scrolling:    disabled
Middle emulation: n/a
Calibration:      identity matrix
Scroll methods:   none
Click methods:    none
Disable-w-typing: n/a
Accel profiles:   n/a

Device:           AT Translated Set 2 keyboard
Kernel:           /dev/input/event0
Group:            6
Seat:             seat0, default
Capabilities:     keyboard 
Tap-to-click:     n/a
Tap-and-drag:     n/a
Tap drag lock:    n/a
Left-handed:      n/a
Nat.scrolling:    n/a
Middle emulation: n/a
Calibration:      n/a
Scroll methods:   none
Click methods:    none
Disable-w-typing: n/a
Accel profiles:   n/a

Device:           SynPS/2 Synaptics TouchPad
Kernel:           /dev/input/event17
Group:            7
Seat:             seat0, default
Size:             53.33x23.57mm
Capabilities:     pointer 
Tap-to-click:     disabled
Tap-and-drag:     enabled
Tap drag lock:    disabled
Left-handed:      disabled
Nat.scrolling:    disabled
Middle emulation: n/a
Calibration:      n/a
Scroll methods:   *two-finger edge 
Click methods:    *button-areas clickfinger 
Disable-w-typing: enabled
Accel profiles:   none

Device:           TPPS/2 IBM TrackPoint
Kernel:           /dev/input/event19
Group:            8
Seat:             seat0, default
Capabilities:     pointer 
Tap-to-click:     n/a
Tap-and-drag:     n/a
Tap drag lock:    n/a
Left-handed:      disabled
Nat.scrolling:    disabled
Middle emulation: disabled
Calibration:      n/a
Scroll methods:   *button
Click methods:    none
Disable-w-typing: n/a
Accel profiles:   flat*adaptive

Device:           ThinkPad Extra Buttons
Kernel:           /dev/input/event7
Group:            9
Seat:             seat0, default
Capabilities:     keyboard 
Tap-to-click:     n/a
Tap-and-drag:     n/a
Tap drag lock:    n/a
Left-handed:      n/a
Nat.scrolling:    n/a
Middle emulation: n/a
Calibration:      n/a
Scroll methods:   none
Click methods:    none
Disable-w-typing: n/a
Accel profiles:   n/a

I have both the the xf86 synaptics and evdev drivers installed.

> pacman -Qi xf86-input-synaptics
Name            : xf86-input-synaptics
Version         : 1.8.3-2
Description     : Synaptics driver for notebook touchpads
Architecture    : x86_64
URL             : http://xorg.freedesktop.org/
Licenses        : MIT
Groups          : xorg-drivers  xorg
Provides        : synaptics
Depends On      : libxtst  mtdev  libevdev
Optional Deps   : None
Required By     : None
Optional For    : None
Conflicts With  : synaptics
Replaces        : synaptics
Installed Size  : 201.00 KiB
Packager        : Andreas Radke <andyrtr@archlinux.org>
Build Date      : Wed 11 Nov 2015 02:01:22 AM MST
Install Date    : Mon 23 Nov 2015 11:50:56 AM MST
Install Reason  : Explicitly installed
Install Script  : No
Validated By    : Signature

> pacman -Qi xf86-input-evdev
Name            : xf86-input-evdev
Version         : 2.10.1-3
Description     : X.org evdev input driver
Architecture    : x86_64
URL             : http://xorg.freedesktop.org/
Licenses        : custom
Groups          : xorg-drivers  xorg
Provides        : xf86-input-driver
Depends On      : glibc  systemd  mtdev  libevdev
Optional Deps   : None
Required By     : xorg-server
Optional For    : None
Conflicts With  : xorg-server<1.18.0  X-ABI-XINPUT_VERSION<22  X-ABI-XINPUT_VERSION>=23
Replaces        : None
Installed Size  : 73.00 KiB
Packager        : Jan de Groot <jgc@archlinux.org>
Build Date      : Fri 08 Jan 2016 05:53:29 AM MST
Install Date    : Fri 08 Jan 2016 11:19:36 PM MST
Install Reason  : Installed as a dependency for another package
Install Script  : No
Validated By    : Signature

Same problem here.
The touchpad configuration was working correctly in 3.18, but since the update in 3.20 in archlinux, it seems unresponsive to settings modifications.

Both 'xf86-input-evdev' and 'xf86-input-synaptics' are installed.
Could this be the cause of the problem ?

Here is the return for 'sudo libinput-list-devices' concerning the device :

Device:           AlpsPS/2 ALPS GlidePoint
Kernel:           /dev/input/event15
Group:            8
Seat:             seat0, default
Size:             71.53x53.33mm
Capabilities:     pointer
Tap-to-click:     disabled
Tap-and-drag:     enabled
Tap drag lock:    disabled
Left-handed:      disabled
Nat.scrolling:    disabled
Middle emulation: enabled
Calibration:      n/a
Scroll methods:   *two-finger edge 
Click methods:    none
Disable-w-typing: enabled
Accel profiles:   none


PS: Actually, even trying to disable the touchpad completely in the control center does nothing. I'm still able to click on tap and move the mouse when the device should be turned off.

Possibly related (or not), but left-right scrolling with two fingers also doesn't seem to work. This did work in 3.18. I don't see a setting exposed for this in the "Mouse & Touchpad" dialog, FWIW.

See bug 764819 for details.

*** This bug has been marked as a duplicate of bug 764257 ***

Thanks! On Arch resolving this was as simple as removing the synaptics and evdev packages in favor of libinput. The "Mouse & Touchpad" settings work as expected so far.