...if the price goes down substantially, to well below that of when I initially invested; is it possible that I might be sent a bill for the further losses ?

No.

If the price of a bitcoin dropped to $0, you would lose your entire investment, but no more than that.

Think of it like buying gold: if the price of gold dropped to zero, you would now just own some worthless metal. You wouldn't owe anyone money, you would simply have lost the capital you used to buy the gold in the first place.

The only way you are going to owe more money than your investment is through some external factor. As an example, if you took out a loan to buy bitcoin, and then owed interest on the loan. But that debt would be owed to whoever lent you the money.

If I invest £2,000 in bitcoin and the price goes down, can I lose more than my investment.

Exchanging Currencies vs Making investments

This is just the same as investing in Euros by exchanging £2000 for Euro banknotes at a bank or money exchange. Once you have those €2227 they can never be worth less than nothing. No Euro banker will come after you and force you to pay them more GBP to take those Euro notes away from you. That's not the way currencies work.

It is more of a currency exchange than an investment. You might exchange GBP for Euros because you need the Euros to buy something (e.g. on holiday in Italy) or because you hope that the exchange-rate will change after Brexit so that your €2227 can be exchanged back for £3000.

It is not like investing in tangible things (e.g. land) and it is not like investing in complex financial instruments (e.g. derivatives of various types) where liabilities can accrue.

From the point of view of someone looking for a store of value, or speculating on exchange rate changes, whether exchanging £2000 for Bitcoin is more like buying Euros or like buying Venezuelan Bolivars is something that only time will tell.

Heh,

You can under extreme circumstances, but margin trading is one avenue where this is possible. Though the exchange is set to liquidate as to not have losses for the margin funding, just your funds unless you set a conservative stop limit. A great example would be the Ethereum flash crash of 2017 on Coinbase, where from a trading price of $340/ETH it dropped to 11 cents and people instantly were liquidated --- though ironically since the exchange liquidated and hovered around 160, most people did not "owe" coinbase much -- and in the end the trades were reversed (after bad publicity.)

Now if you use bitmex, you'll lose your investment on the slightest of market movement but as for losing your house. Unlikely.