Q:

Species identification: what kind of fungus is this?

Found in a cave
Location: Israel
The white tentacles appear to be sticky. When they touch something, they stick to it and detach from the main body

A:

I think these are parasitized insects. 
Fungi of various types can infect insects.  The end stage produces these fruiting bodies.  Here is an example I found on the excellent site WhatsThatBug.

Here is another example.

http://www.anura.it/the-parasites-parasite-everyone-has-a-bad-day/clasp04/
The second site notes

Entomopathogenic fungus (Clavicipitaceae sp.) killed a troglophile
  moth, Veins cave (Grotta delle Vene), Italy

I would guess something similar for what you found. 
I must say your images are incredible.  The long fungal tendrils on the second image are the most amazing example of this I have ever seen.

