Let's start with some basics - You as the client have the absolute right to terminate the lawyer's services at any time. (Fracasse v. Brent (1972) 6 Cal.3d 784, 790.) When a client discharges a lawyer, the lawyer is entitled to be paid for the services provided as set by the fee agreement.
An attorney, however, does not have an absolute right to withdraw from representing a client for any reason or whenever he or she feels like it. The way an attorney withdraws, and under what circumstances, can affect whether the attorney can enforce a lien against the client's case. For example, an attorney may not withdraw until the attorney has taken "reasonable steps to avoid reasonably foreseeable prejudice to the rights of the client, such as giving the client sufficient notice to permit the client to retain other counsel. (Rules of Professional Conduct, Rule 1.16(d).) If I understand your question correctly, your attorney quit in the midst of your deposition, which could be highly improper if the deposition is continuing immediately.
Under certain circumstances, attorneys must withdraw from representing a client, such as when a client asks the attorney to violate the law or rules of professional conduct, or where the attorney has a mental, physical, or emotional condition that makes it unreasonably difficult to continue the representation. Attorneys must seek a court order allowing their withdrawal.
In other cases, an attorney may permissively withdraw. The most common scenario is a personality clash with the client. This is good cause for withdrawal, but it is not necessarily a justifiable excuse that would allow an attorney to enforce a lien. (Estate of Falco (1987) 188 Cal.App.3d 1004, 1014). And, again, the attorney must get court approval.
The circumstances leading to your attorney's withdrawal are not fully described, but what is described seems suspect. Request your complete client file, both hard copies and electronic documents. You are entitled to everything at no cost to you, even if the attorney says you owe him or her money. And, then seek in-person advice right away. If your deposition is set to continue, ask the other attorney(s) involved to "stipulate to a continuance of the deposition," and put the agreement to postpone your deposition on the record with the court reporter in the room....
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