LONDON, Oct 5 (Reuters) - Business growth across Europe remained strong last month but elevated inflationary pressures put a dent in demand while ongoing supply issues constrained activity, issues which are likely to continue, a survey showed on Tuesday. Although many restrictions imposed to contain the coronavirus pandemic have now been lifted in the region, firms are suffering from shortages of staff, raw materials and transport. HIS Markit"s final composite Purchasing Managers" Index (PMI), seen as a good guide to economic health, sank to 56.2 last month from August"s 59.0, although still well above the 50 mark separating growth from contraction and just above a 56.1 "flash" estimate. On Friday, a euro zone manufacturing PMI showed growth remained robust in September but activity suffered from supply chain bottlenecks, and the bloc"s dominant service industry also saw the pace of expansion slow. A PMI for the services sector fell to 56.4 from 59.0, its lowest since May, while the new business index dropped to 55.3 from 57.9. "Today"s data confirm the services sector remains on the path of recovery, but the pace of expansion softened. Despite the momentum remaining solid, inflationary pressures and supply-side disruptions weigh on the outlook," said Maddalena Martini at Oxford Economics. "Looking forward, we see these downside risks to remain towards the end of 2021." Demand fell to a five-month low as firms passed on part of rising input costs, which rose at a record pace, to consumers. The composite output prices index rose to 59.1 from 58.3, not far from survey highs set in the summer months. Activity in Germany"s services industry continued to grow strongly in September, but the recovery from the pandemic lost momentum as catch-up effects waned and more companies were affected by supply bottlenecks. read more In France, services growth slipped as inflationary pressures and COVID-19 protocols had an impact on businesses. Italian and Spanish services growth also slowed. Meanwhile, in Britain - outside the currency union - the post-lockdown economic recovery avoided losing further momentum last month but companies increased prices at the fastest pace on record, adding to signs of rising inflation.