A root meaning “go, proceed” appearing as one of the many etymologies of Q. vanwa “gone, departed”. It may have been a partial restoration of the (rejected) meaning of the root ᴹ√AB “go away, depart” from The Etymologies of the 1930s, a root that later was altered in that document to mean “refuse, deny, say no” (Ety/AB). It may also have been intended as a replacement for ᴹ√WAN “depart, go away, disappear, vanish”, also from The Etymologies and the basis for ᴹQ. vanwa in the 1930s.
In the published corpus, the first clear mention of this root is in a discussion of the Quenya future suffix -(u)va from the late 1940s where it was given as √BĀ, BANA “go, proceed” with Q. vanwa “gone, over” as one of its derivatives (PE22/97). It appeared again in a discussion of the Sindarin greeting mae govannen “well met” from the late 1950s, where Tolkien hypothesized that the verb “meet” might be go- “together” + ban- “go” or “meet”, which he again connected to Q. vanwa “gone” (PE17/16). Phonetic complications prompted Tolkien to coin a new verb covad- “assemble, gather together” from a root √KOB, but he was unwilling to commit at this time to the new verb because of the obvious similarity of govannen to √BAN “go” (PE17/16-17).
However, in 1959 when he was overhauling the Eldarin system of negation, it seems Tolkien finally rejected √BA(N) “go” because of the conflict with √BĀ/ABA “refuse”, and he transferred the sense “go” to the root √MEN (PE17/143). In etymological notes contemporaneous with the Quendi and Eldar essay of 1959-60, Tolkien restored √KOB “gather” (PE17/150) though he seems to have eventually settled on the form √KOM for this root along with a restored verb S. covad- “bring together, make meet” (PE17/157-158). Q. vanwa “gone” was also transferred to √WĀ/AWA by the time of the Quendi and Eldar essay (WJ/366). Meanwhile, the Quenya future had been transferred from √BA to the root √UB in the early 1950s (PE22/132).
This left no remaining functions for √BA(N) “go”, and there is no further sign of it starting in the 1960s.
The root √BAN (or some variant) was connected to beauty for most of Tolkien’s life. One notable derivative was the name of the Valie Vána. In the Qenya Lexicon of the 1910s Tolkien gave this root as ᴱ√VANA (unglossed) with derivatives like ᴱQ. vane “fair, lovely” and ᴱQ. vanesse “beauty”, but also ᴱQ. Vanar as another name for the Valar (QL/99). The Gnomish words had distinct forms like G. Ban “Vala” (GL/18) vs. G. gwant “beautiful” (GL/44). Thus it seems that ᴱ√VANA in the 1910s is actually a blending of two roots, ✱ᴱ√BANA (or ᴱ√ɃANA) “divine” vs. ✱ᴱ√WANA “beautiful”. This second variant reappeared much later in the 1959 root √(G)WAN “fair, pale” (PE17/154; WJ/383); see below.
In The Etymologies of the 1930s Tolkien gave this root as ᴹ√BAN and it seems at this stage it was associated exclusively with beauty, given derivatives like ᴹQ. vanya and N. bein “beautiful, fair” (Ety/BAN). These words survived into later iterations of the languages as Q. vanya and S. bain (PE17/150). The root √BAN reappeared with these derivatives in several etymological notes written around 1959 (PE17/149-150, 165).
In these 1959 notes, Tolkien first connected √BAN to “fair” as it was in the 1930s (PE17/149), but reversed himself, noting “VAN cannot only = fair (blonde), since vanima is applied in LR to Arwen who was like Lúthien dark” (PE17/165). This prompted Tolkien to reintroduce the root √(G)WAN “fair”, as noted above, limiting √BAN to “beauty” only. He then refined its meaning, saying “√BAN ... appears originally to have referred simply to ‘beauty’, but with implication that it was due to lack of fault or blemish” (PE17/150). The root applied to Vána because she was perfect and unmarred in her beauty.