7a.transitive. To lash (a person) with or as though with a whip or cord; to beat, thrash.
1599 H. Porter Pleasant Hist. Two Angrie Women of Abington sig. I4v I doe not loue to be lac't in, when I goe to lase a rascall.
c.intransitive. Originally and chiefly U.S.to lace into: to attack or assault, esp. verbally; to criticize strongly; to lay into.
1908 Black Cat Oct. 17 She laced into me with all kinds of abuse.
Etymology of "lay":
In sense 7 apparently by association with lash v.1
32d.to lay into: to belabour; to ‘pitch into’. slang or colloquial.
1838 D. Jerrold J. Applejohn in Men of Char. xiii I shall be very happy..to go and hold the door, while you lay into the ruffian.
70 years between the two first citations is not much when you consider that they are just the first written example anyone has found so far. But yes, "apparently by association with lash" is the OED's answer to your question.