There is no overall shortage of prescription medicines, but there has been an increase in demand for all types of products - particularly hand sanitisers, paracetamol and thermometers.
Mark Lyonette, chief executive of the National Pharmacy Association (NPA), which represents independent community pharmacy professionals, said providers were doing everything they could to maintain the supply of medicines.
The NPA warned that the sharp surge in demand for medicines could lead to longer waiting times while pharmacy staff process prescriptions.
"In most cases, pharmacies are currently able to meet need in a timely fashion, even if that means limiting the quantity of certain medicines sold to each customer," Mr Lyonette said.
"Certain products, for example, hand sanitisers, paracetamol and thermometers, are only intermittently available.
"Please only buy the medicines you need now for you and your family; this will help to avoid creating difficulties for others, so that everyone in your community gets the medicines they need."
Speaking to Sky News, Dr Clare Gerada, former chair of the Royal College of GPs, asked the public: "Please do not ask for early prescriptions.
"If everybody is going to ask for two months on top of the two months they've already got - we're going to run out.
"We won't be issuing early prescriptions because we don't want any stockpiling".
Meanwhile, online pharmacies have begun limiting orders as they struggle to cope with an "unprecedented" demand for their products.
It asked customers to buy products "only as and when needed" to "ensure pharmacies can continue to provide care for everyone over the coming months".
ChemistDirect said on its website that orders would take between five and seven working days to dispatch due to "exceptionally high demand".
Similarly, Pharmacy First informed customers that it was receiving an "unprecedented amount of orders" and that there were currently delays of up to seven to 10 days to process orders, asking customers to "only order what they need".
The chief executive of the General Pharmaceutical Council said that pharmacy professionals were working under "significant pressure" during the coronavirus crisis.
Duncan Rudkin said there had been increasing numbers of reports of pharmacy staff experiencing "abuse, disorder and even violence" from members of the public.
"Abuse of pharmacy staff is never acceptable," he said in a statement.
"Pharmacy is a key part of the national response and its workforce are entitled to be treated with the same respect as other key healthcare professionals.
"This is an especially challenging time for the pharmacy profession and we condemn any abuse."