Let me answer your questions in reverse order:
...or is fractional reserve banking by individual corporations more to blame?
The majority of money created in the US (and indeed, the world) is created by private banks through fractional reserve lending, not by central banks creating fiat money.
Monetary base refers to money that banks hold in their accounts at the federal reserve as well as currency in circulation -- in other words the money that the government has the most control over.
M2 is a measure of money that includes bank lending. Although the ratio of M2:BASE has fallen in the last few years, M2 is still clearly far bigger than BASE.
Now back to the first part of your question:
is there any proven relationship between unbacked currency and extreme inflation
Depends what you mean. Unbacked currency doesn't lead to runaway inflation in all cases. Look at all of the major currencies today: slow and steady inflation for most of the last 100 years.
In a few cases, unbacked currency has been involved in runaway inflation. Typically runaway inflation is not strictly a monetary phenomenon, but also involves political factors. (See Zimbabwe). I wouldn't call this "proven", however.
There are also some instances of hard monies experiencing dramatic inflation throughout history. European monarchs accomplished this by debasing pure gold and pure silver coins with cheaper metal alloys. The US had a number of financial panics related to rapid increase and then collapse of money supply even when on a supposed "gold standard", e.g. the Panic of 1819.
The Panic of 1819 is a controversial topic. Although it does indicate short comings of a "gold standard", it is essential to remember that the government was still manipulating the money supply even in 1819, and therefore the gold-backed money was still not a free market money.
I'm not aware of any cases in history where a commodity money or commodity-backed money experienced hyperinflation on the scale of Zimbabwe.