You possibly have this already, a newspaper item mentioning a Job PETERS at Lyndhurst in 1887, not that far from Lymington.
"LYNDHURST PETTY SESSIONS ...
DAMAGE TO TREES.- Job Peters and George Lakey, gipsies, were summoned for willfully damaging trees at Minesteed, on the12th ultimo. by chopping off branches, to which they pleaded gulity, saying they thought there was no harm in chopping off a little dead wood, but would take care they did not appear again. - The damage was estimated at 6d, and the value of the wood at 1d. - The defendants were ordered to pay the damage. a fine of 5s and 5s 5d costs. - Lakey did not pay the money, and was committed for fourteen days with hard labour. ..."
Saturday, Mar. 12, 1887, Southampton Herald