Can stress or training influence the perception of time? I have had this experience that I fell with my bike on an icy street. During the fall, time seemed to slow down and I had an apparent age-long time window to stretch my hand and safely catch my fall.

A less anecdotal example is the following. Among jugglers I think it is a common phenomenon that once you get the hang of a trick, that at that point one comes to realize 'how much time one actually has to catch a ball and throw it back into the air'. (I looked for references, but it at least applies to me, and I remember to have read it in juggling instruction material as well. I can try to find more evidence of this perceived time when needed).

Both examples show that the perceived lengthening of time appear to allow the body to make more coordinated movements within a given amount of physical time.

Now my question is, assuming physical time is constant (barred Einstein's relativity of time when travelling at great speeds):


  Is it true that perceived time can change under the influence of stress or due to training?
Part of the difficulty in studying time perception is that memory is known to be biased by numerous factors including arousal and salience.  So while people commonly report time slowing down during specific events, it is difficult to differentiate the effects of retroactive memory bias in encoding and recall from actual increased resolution in the perception of time and accelerated physical abilities or mental processes.  For example, adrenaline is known to affect both physical response and memory consolidation.

One hint regarding the plasticity of the brain's actual perception of time comes from the temporal binding effect (nice review here).  A variety of experiments show that the brain automatically recalibrates simultaneity amongst different neural processes through learning: That is, the brain processes different stimuli at different rates, but it learns to calibrate those processes so as to create the illusion that they occur at the same time:


  At least over a short time frame, our temporal perception of events is
  far from veridical, and multiple timelines are capable of dynamic
  recalibration.


In other words, there is no single perception of time, time is perceived at different rates by different parts of the brain, and then retroactively synchronized.

A more direct attempt to differentiate memory from perception was done by Stetson et al (2007).  In this experiment, subjects were dropped 150 feet with no ropes or parachute - into a safety net.  During the free-fall, they were asked to read numbers flashed on a wrist-watch.  The numbers were flashed at different rates to test whether perception resolution was actually enhanced during the fall.  Despite estimating their fall to take much longer than control subjects not performing this task (corroborating the effect of salience on memory), there was no evidence of increased time resolution:


  Our findings suggest that time-slowing is a function of recollection,
  not perception: a richer encoding of memory may cause a salient event
  to appear, retrospectively, as though it lasted longer.


However, this conclusion has been criticized by other authors as the failure to produce an effect on time perception may be due to the experiment not being frightening enough, for only testing visual perception, and that free-fall may actually distract from the reading task.  Evidence from life-threatening events such as accidents and the performance of skilled individuals in extreme sports for example, suggests that increased time resolution, physical performance, and cognitive abilities during extremely stressful events may be possible:


  This evidence supports the suggestion by Arstila (2012) that humans
  possess a specialized hormonal or neurophysiological mechanism for
  high-speed cognition; and that this is activated inadvertently by real
  fear of imminent violent death in accidents, emergencies, and certain
  extreme sports; and may on some occasions be activated intentionally
  by individuals who have trained themselves to do so.


Edit (re updated question):

The latter paper - Buckley (2014) - summarizes research on the performance of experienced / trained / skilled individuals.  Accelerated performance (physical and mental) is found in accidents in extreme sports, such as climbers in accidental falls; and in the performance of many physical activities:


  Less skilled participants perceive only confusion, and are likely to
  freeze, panic, or act in ways which increase danger (Buckley, 2012).
  It is also experience and training which allow skilled exponents of
  many physical arts to achieve feats which appear impossibly fast and
  precise. There are many examples in ball and boardsports, gymnastics,
  acrobatics, dance, martial arts, archery, shooting, swordsmanship, and
  in aircraft, car and motorbike racing and stunt driving.