1877 was one of the few years in which no farthings were issued for circulation in Great Britain, though a small number of proof coins were struck. Freeman catalogued the 1877 proof farthing as Freeman 535 and noted that they were struck some time after 1877 (p137, Freeman, The Bronze Coinage of Great Britain (1860-1970), 2016), though it is not clear when exactly. Peck catalogued the 1877 proof farthing as Peck 1895 and noted that it was struck some time after 1877. Peck had an example in his collection and had examined at least three other examples (p443, Peck, Englsh Copper, TIn and Bronze Coins in the British Museum, 1558-1958, 1970), suggesting at least four proofs were struck.
An example was offered for sale in 2015 in Colin Cooke's farthing list 81 as lot 175. It was offered for £8,750 with the provenance of being from the Slaney Collection, and previously from the Lord Grantley collection. Presumably this was one of the coins that Peck examined when writing his book.
More details about British farthing varieties can be found in A Treatise on Varieties of British Fractional Bronze 1860-1970.