Where does the phrase "on the lam" come from?

Origin:

Lam is by definition to leave or flee, especially from the law.
The confusion about this phrase is that lam is not a commonly used word. Many people assume the phrase to be on the lamb.

From http://www.english-for-students.com/On-The-Lam.html

The term came from 1682 when a group of Quakers were going to be arrested on their flight to America so instead of taking their group along a road they had the ship pick them up in the middle of the night to escape from the Red Coats and The Church of England. The ships name was the Lamb. This ship was part of the William Penn's flotilla. The group that was on the Lamb was headed by Cutberth Hayhurst, his wife and kids, his brother and wife and kids, and his sister and husband, along with a few others. These folks were my forefathers. There is plenty of info on the ship called the Lamb, their escape, and the Hayhurst's. Hence the term on The Lamb.

Nick Hayhurst

When i think of the term "On the lamb", i usually associate it with American gangsters during the depression and 1950's. Therefore i'm more inclined to go with Herman Lamm. Herman k. Lamm was a german born bankrobber who lived between 1890 and 1930. He is considered to be the father of modernday bankrobbing.

https://en.wikipedia.org/wiki/Herman_Lamm

This question was posted in 2011, but apparently there had been studies on the etymology of this term that haven't been discussed in existing answers. There is a 1998 article on this exact topic in The New York Times Magazine: On Language; On the Lam, Who Made Thee? By WILLIAM SAFIRE, MARCH 1, 1998:

In The Random House Historical Dictionary of American Slang, J.E. Lighter defines the term as prison lingo for ''an act of running or flight, esp. a dash to escape from custody.'' In his 1886 ''30 Years a Detective,'' Allan Pinkerton, the first ''private eye,'' explains an operation of pickpockets: ''After he secures the wallet, he will utter the word 'lam!' This means to let the man go and to get out of the way as soon as possible.'' Lighter cites do a lam, make a lam and take a lam early in this century, finally emerging as the passive state of being on the lam.

And the OED's information on its Scandinavian origin is echoed here:

Lighter speculates that it may be rooted in the dialect Scandinavian verb lam, as in the 1525 ''his wife sore lamming him,'' meaning ''to beat, pound or strike.'' Mark Twain used it twice: ''lamming the lady'' in 1855 and ''lam like all creation'' in 1865, both clearly meaning ''to beat.'' The suggested connection is that to avoid a feared lamming (related to slamming), one lams.

So this theory speculates that there's the verb lam first, attested by Mark Twain's use of the word in his books. Then possibly a new meaning evolved out of the verb: in order to not get lammed, one goes on the lam.

Other theories also exist:

At the University of Missouri at Rolla, Gerald Cohen, a professor of foreign languages currently at work on a slang dictionary, has another theory. He notes the cant lammas in Eric Partridge's Dictionary of the Underworld, the lingo of costermongers in London around 1855, alternatively spelled nammou, meaning ''to depart, esp. furtively'' and related to vamoose in the lingo of the American West.

''Namase with its variant spellings,'' Cohen says, ''was the standard cant term for 'leave/make off/depart/skedaddle.' I don't know why nam became lam, but the meanings are the same.''

Twenty-three ships carried William Penn and his followers to Pennsylvania in 1681 and 1682, one of them being the Lamb, which sailed from Liverpool and arrived on 22 October 1682 (“William Penn's Ships – 1681/2, https://www.chester.pa-roots.com/brittish_research/william_penns_ships.htm).

That does not necessarily mean, however, that the noun lam, as in be on the lam, comes from the name of the ship.

After puzzling over this phrase for quite some time I came across the possible answers offered in this site. The one that seemed to show the most promising solution was the longest entry, which included the possibility of there being a verb “to lam” meaning “to run.”

I found just such a verb in the English Dialect Dictionary, a somewhat now forgotten, yet very useful, text. I hereby provide the germane entry, from Vol. III, pp. 509-10:

LAM...4. to run quickly. w. Yks 5 Whear’s tuh lamming tul! Ther wur a peeler after him - by Gow didn’t he lam!

The above quotation - which seems to fit exactly the meaning of “lam” - as “running from the police” (i.e. a “peeler”) - comes from C.C Robinson’s The Dialect of Leeds and its Neighbourhood, of 1861.