I am asking a solicitor for a copy of an electronically signed contract. They keep telling me that'no physical copy exists'. I am sure there must be some form of a copy which can be produced otherwise how do they prove the contract exists?
how do they prove the contract exists?
By producing --in court-- the exact same contract they are unreasonably declining to produce per your request.
Their pretext that "no physical copy exists" is devoid of merit. They should still be able to send you electronically the contract you signed electronically. If they are genuinely unable to do so, then they should have no objection to rescind a contract they supposedly can no longer identify.
A refusal to give you a copy might be tantamount to constructive breach of contract insofar as they are trying to keep you ignorant (in the sense of unaware, not of reckless or blatant) of the conditions you knowingly and willfully entered when the contract was formed. The party who does not know/recall those conditions because he is unreasonably prevented from knowing/recalling them cannot be expected to perform under the terms of that contract. In that case, the refusing party risks forfeiting enforceability of his rights & remedies pursuant to the contract.
If your motivation for requesting a copy is to know your rights (rather than your duties), then you might want to seek injunctive relief via court proceedings to the effect of compelling the counterparty to produce that contract.