It may sound a little absurd to say this, but my feeling is that there are two different kinds of ‘me’. I attempted to grapple with this issue on a thread titled something like ‘What do we mean by ‘self’?’
The ‘me’ that Krishnamurti talks about is the psychological ‘me’, the ego, the observer. This ‘me’ is the creation of thought, of the accumulation of memory. This ‘me’ of memory and thought is the time which separates us from the actuality of the Now moment. When the observer is the observed - in the fullest meaning of this phrase - psychologically speaking, the ‘me’ comes to an end. This is what I understand Krishnamurti to be talking about.
However, distinct from this psychological ‘me’, there is also the functional centre of the brain-body. This functional centre is not a spiritual centre, nor is it a psychological centre. It has no self-consciousness at all. It is simply the nervous system and neurological processing of the organism for purposes of physical survival. Krishnamurti sometimes refers to this as the ‘body’s own intelligence’. It is why our bodies are proprioceptive, capable of distinguishing inner from outer, light from dark, hot from cold, the various sensations, subject and object (at the level of organic perception).
So the space between the body and a tree, the distinction of inner and outer physical space, I would suggest occurs because of this functional centre. This functional centre ceases to exist at physical death, and is necessary for physical survival.
But the psychological self, the psychological ‘me’, which also creates space between myself and the tree (a psychological space), and which creates a psychological division between inner and outer, is not necessary to the survival of the body, and can come to an end through insight - such as the insight that the observer is the observed.
Then, as you say, one may be
I hope this makes some sense to you.