Dutch attorneys are not allowed to take cases on a contingency basis. 
I'm looking for an American, Canadian, or British Law Firm (which are
able to take contingency cases) with an office in The Netherlands. 
The case is a malpractice case against a Dutch attorney, but must be
handled on a contingency basis

I suspect that even non-Dutch attorneys in the Netherlands will also
have to comply with the rule of not taking contingency work.

Here in the UK, all professionals carry Indemnity Insurance which
makes it almost impossible to bring a malpractice case. Could be the
same in NL!

I agree. If there is such a regulation (and there is - no contingency
basis payment in Legal Aid cases), it applies to anyone who has a
license to practice law in the Netherlands, and it doesn't matter if
they are part of a foreign law firm operating in the Netherlands, or a
local Dutch firm - a multi-national firm is not above the jurisdiction
of each country (or so should it be, and so it is in any place where
there's rule of law) where it operates, and must apply to the same
regulations.

According to a brief (and not very comprehensive) Contingency fee
agreements are prohibited or limited in most countries in the EU and
in any case "no jurisdiction within Europe allows contingency fee
agreements of the type available in the US." (SOURCE: "Mobile Phone
Litigation: A US Phenomenon?"
<http://www.ashursts.com/pubs/pdf/2122.pdf>).

My suggestions: 

- It really depends on your case (torts?), but I could suggest firms
specialising in torts (if that is the nature of the problem) in the
Netherlands. I am sure that a good Dutch lawyer would give you the
best advice possible within the limits of the Dutch law.

- To reconsider your strategy regarding the whole matter. I hope I
don't sound too crude, but when dealing in and with different
cultures, don't try to impose your (legal) culture, but to understand
the other culture. When in Rome, do as the Romans do (eat pasta).

Try www.martindale.com to search for firms in the Netherlands and in
any country, actually.  Also visit your local library to see if the
reference librarian can refer you to the printed version of
international law directories. Without giving names, can you give a
brief description of exactly what type of malpractice occurred or what
type of case you were pursuing or exactly how much in financial or
personal damages you suffered and what you expect the remedy to be?

What I meant to say was that if you have access to a library that has
the printed book version of Martindale Hubbell Law Directory, that you
could look at that book to find firms located in the Netherlands and
then contact each one by e-mail to find out whether they would be
sympathetic to handling your case.  In view of the fact that they
don't do contingency fee based cases, you should consider avoiding
that phrase and wording your fee request in another way, perhaps
requesting fee quotations/estimates, or by explaining that can't
afford an exorbitant fee amount but would be willing to reach a fairly
negotiated price--if you are lower income please mention that fact so
they will be sympathetic to your situation.

4KEITH (I'm NOT a GOOGLE Researcher)