Laudator Temporis Acti

James Mitchell, Significant Etymology, or Roots, Stems, and Branches of the English Language (Edinburgh: William Blackwood, 1908), p. 59:

The word branch certainly was derived from the F. branche (in Breton, branca), as in Italy and Spain, and there can be little doubt, I think, that the F. branche was derived from the L. brachium, the arm. But a direct derivation from brachium is inadmissible. It is necessary for this to have had a L. form brancia.

There is considerable doubt about a connection between branch and brachium. See, e.g., The Concise Oxford Dictionary of English Etymology (Oxford: Oxford University Press, 1986; rpt. 1996), s.v. branch (p. 48): "XIII - (O)F. branche :- late L. branca, of unkn. orig."

For some interesting remarks on the root of branch, see Robert S.P. Beekes, "The Etymology of Dutch Broek 'Breeches'," in Dirk Boutkan and Arend Quak, edd., Language Contact: Substratum, Superstratum, Adstratum in Germanic Languages (Amsterdam: Rodopi, 2000 = Amsterdamer Beiträge zur älteren Germanistik, 54), pp. 25-26 (at 26):

When I looked at the etymology of French branche 'branch' (the English word is a loan from French), which is derived from (Lat.) branca, it occurred to me that the form of this word could be a variant of the word for 'breeches', and that the meaning could be compatible too. The word branca means 'paw' (see Ernout-Meillet). Thus also in Rumanian (brînca). See Pfeifer on Germ. Pranke 'Klaue, Hand, Pfote, Tatze', which is a loan from a Romance language. For the meaning, cf. Russ. noga ('nail, claw > foot, leg'). (In French it developed further into 'branch', which will have gone through 'leg'.)

Pfeifer is Wolfgang Pfeifer, Etymologisches Wörterbuch des Deutschen (Berlin: Akademie-Verlag, 1989), unavailable to me. Ernout-Meillet is their Dictionnaire étymologique de la langue latine (Paris: C. Klincksieck, 1951), s.v. branca, -ae f.:

patte. Mot très rare et tardif; Gromatici (deux exemples), Aug., Serm. (un exemple). M.L. 1271 (fr. branche). Passé en germ. branka "Pranke" et en irl. braice. Mot gaulois?

The two examples from the Gromatici can both be found on p. 309 of Carl Lachmann, ed., Die Schriften der römischen Feldmesser: Gromatici Veteres (Berlin: Reimer, 1848):

Terminus siue petra naturalis si branca lupi habuerit, (fig. 249.) arborem peregrinam significat.

Terminus siue petra naturalis si branca ursi habuerit facta, (fig. 250.) lucum significat.

Unfortunately, I don't have access to Brian Campbell, The Writings of the Roman Land Surveyors: Introduction, Text, Translation and Commentary (London: Society for the Promotion of Roman Studies, 2000; rpt. 2008 = Journal of Roman Studies Monograph, 9).

Pierre-Henri Billy, Thesaurus Linguae Gallicae (Hildesheim: G. Olms, 1993), p. 33, gives the Augustine citation as "Aug., Serm., CLXI, 4," but I can't find the word in that sermon.

Ernout-Meillet's "M.L. 1271" is a reference to Wilhelm Meyer-Lübke, Romanisches etymologisches Wörterbuch (Heidelberg: Winter, 1935), #1271 (unavailable to me).

Two older conjectures about the root of branch have not met with favor. F. Neumann, "Französische Etymologien," Zeitschrift für romanische Philologie 5 (1881) 385-386 (at 386), speculated that branca came from Latin bi-ramica, i.e. from the roots bis (twice) and ramus (branch), and C. Nigra, "Note etimologiche e lessicali, terza serie," Archivio Glottologico Italiano 15 (1901) 97-128 (at 100-101), derived branca from Germanic *krampa. According to Le Trésor de la Langue Française informatisé, s.v. branche, these conjectures "sont à écarter" (are to be avoided"), and Jean R. Scheidegger, Le Roman de Renart, ou Le Texte de la Dérision (Genève: Droz, 1989), p. 100, describes them as "peu vraisemblables" (not at all likely).

Finally, Anatoly Liberman, Break - Broke - Broken, writes:

Words designating all kinds of fragile things tend to be related to break. Among them is probably bracken "fern," borrowed, in all likelihood, from Scandinavian. Its cognates have been attested with the senses "branch," "bush," "juniper," and so forth. The initial meaning of bracken must have been "tree" or "brushwood." People broke trees and called the product brak-. Later some of those words acquired more specialized meanings, "fern" and "juniper," among others. The ties between Latin branca, from which, via French, English has branch, and brak- cannot be direct, for Latin cognates of break begin with fr- (compare fragile, a Romance word), but the coincidence is curious.