DUBLIN, Nov 3 (Reuters) - - Growth in Ireland's services sector fell to a fresh 3-1/2-year low in October as Britain's decision to leave the European Union continued to weigh on new orders, a survey showed on Thursday.

The Investec Purchasing Managers' Index (PMI) of activity in services fell to 54.6 in October from 56.2 in September, the lowest reading since May 2013, when Dublin was still working its way through an international bailout.

The index, which covers companies from banks to hotels, has fallen from a 10-year high of 64.0 in January as new business has steadily declined.

Investec Ireland chief economist Philip O'Sullivan said the fact new business had fallen for five consecutive months was a "particular concern", with a number of panelists citing the Brexit vote and related sterling weakness as a factor.

Ireland is widely considered the European Union economy to have the most to lose from the June referendum vote in its key trading partner.

But respondents remained optimistic about the future, with 49 percent predicting activity would increase over the coming 12 months compared to 7 percent expecting a decrease.

The rate of employment growth in the sector improved slightly, with more than 14 percent of respondents taking on extra staff during the month, while 6 percent cut jobs.

(Reporting by Conor Humphries; Editing by Catherine Evans)