It is current that kills you, one way or the other. Either via simple interference with the electric impulses that keep your heart and brain working, or through a combination of that and heating effects.
Whilst we all know that you need voltage to drive current, this is simply down to impedance of the load. In this case the load is your body, so the voltage required to drive that current will depend on the impedance of the body in that situation.
There are many different factors, including whether it's AC or DC you come into contact with, the path through your body, the resistance of that path and the length of time you are in contact.
It's generally taken that anything above 30mA AC, via a path across the heart, is in the 'possible death via cardiac arrest' category. Higher currents will also cause tissue damage (via I2R heating effects) and increase the chance of a fatality. 30mA will be familiar to some as it's the level at which most RCD's operate, for exactly this reason. Also, fault disconnection times are based on reducing the contact time to lower the chance of fatality. Humans can generally perceive shocks of 1mA and upwards.
The resistance of the human body is not a fixed value. It varies by tissue type and from person to person. Skin resistance is quite high (1k to 100k). The internal resistance of the body can be as low as 300-400 Ohms. In high voltage incidents the usual sequence is that the skin will be charred, via burning effects, thus lowering resistance substantially as it breaks down.
We do occasionally have fatalities in the power industry, but I also know people that have come into contact with 11kV and been lucky enough to live. One chap re-energised a circuit and got distracted. He then happened into a substation and grabbed hold of some test prods to remove them, right across phase-to-phase 11kV. Another chap climbed a pole and came in contact with the top side of a pole stay, where the insulator had failed, coming into contact with phase-to-earth voltage of 6.35kV. Conversely I know of another guy that died when an overhead line insulator failed during 11kV switching.