Live Looping Audio On The Fly?

is it possible to live loop audio in Ableton like I do with my Boss RC600 looper pedal. For example, I will repeat a guitar phrase until it sounds as near perfect as my musicianship allows, then using the footswitches with sharp timing I will create a loop of this phrase. I then stop recording and listen to the loop while searching for harmonisations, and when again I find and refine one that sounds good I again use the foot pedals to layer this harmony onto the original loop, and so on until I have a multi-layered sound, all done on the fly as I continually play.
Is Ableton even designed to do this? I am new to Ableton, having just acquired both Ableton Suite and a Push 2. I did wonder whether live-looping would be possible using a footswitch into the Push 2, but I note that the footswitch is a bit slow in triggering Ableton, not the instantaneous action I need and that was on my looper pedal. Perhaps this is because I am using a piano pedal, so is there perhaps another type of footpedal that would work for this? Or would this work with a dedicated foot controller for Ableton? Or could I even control Ableton from the footswitches on my looper pedal? Or is there some other possibility?
Many thanks for your help.