Whilst accreditation is important for a laboratory carrying out a test, it does not mean that a lab cannot do it if it is not accredited.  In this instance I would think the minimum requirement is to look at their performance in a proficiency testing scheme for the assay you want them to do.  If they are not in a PT scheme or one is not available (are they in any PT schemes?), then ask to see their internal quality assurance (IQC) performance for the test.  If they have no IQC system in place then clearly they do not offer sufficient quality assurance for you to have confidence in their result.  Perhaps they could set something up if it is an unusual assay, but in general a lab with no quality management system in place cannot be much use to an external customer.

Kind regards

as a principle, the need for an accreditated lab (according to ISO 17025, GLP, etc.) is MANDATORY when the results are to be used in regulatory assessments - therefore, a lab which is capable to reliably perform a test (i.e., its results have known accuracy and precision, they are affected with known uncertainty, etc.) can do the job  without any accreditation, but you have to spent work (a lot, indeed) to achieve confidence in the results you receive for each single test you require. Accreditation implies that such assessment has been already performed (and documented, audited, continuously evaluated by means of proficiency tests, etc.) by a third party accreditation Body. 

Please, note that a laboratory is accreditated for a known and documented number of essays, but this DO NOT exclude its capability to perform other essays, even if not accreditated. Remember: the right expression is: 

LabX is accreditated for the following essays: iron, chromium, ..., in drinking water by ICP-MS in the following concentration range: ...ppm to ...ppm 

the expression LabX is accreditated has no significance 

There is no requirement unless you are making a measurement for a specific purpose that has either a standard established or a regulatory requirement. A specific purpose is one that has an ultimate legal responsibility or an experimental requirement for publishing under specific guidelines. An experimental measurement has only the requirement that it is defensible in the scientific community.

A defensible measurement is one performed under reproducible conditions. A good measurement with good confidence may require that equipment or standards have calibrations from NIST or other similar agency. A defensible measurement is one where errors are estimated and propagated to the final, reported value

I have the opinion like a person who acquired experience in the lab in one National Metrology Laboratory that the knowledge of metrology and their basic principles can do a lot for you in the lab. The first question is even the nature of the property or of the method, is the result dependent on the method? Can you trace back the results to SI units and their measurement standards? Can/are the results traceable to the same reference and uncertainties of the method properly estimated? Is the method validated? Is the person technically competent to do that very specific assay in the laboratory with proper QA&QC? "That is supposed to be covered by accreditation", as very properly mentioned by Antonino Maggio. Only a reliable PT, not any PT can be necessarily reliable, or equivalent comparison let you know who is rigth. I would recommend to perform  a comparison at least as excersice between you. Accreditation is valuable as formal for many official purposes, but what really matters is the true and proved quality of the results produced by the person in the lab conditions using that specific method. There is no other way to do it easily do a comparison. Carry out an audit according to ISO17025 for the technical elements. Non standardized methods need to be validated at least in their relevant performance characteristics.

Be very careful with concepts even scientific comunities if then want to compare values of measurements results scientifically sound that are very significant for decisions they need to be traceable. Precision is a very old fashioned concept and only a part of the uncertainty statement, you also need to know the accuracy and be able to have credible measurements uncertainties, and for this that exists the measurement standards. They have values, or is some cases scales, for example  like calibrated mass pieces, to which compare your measurement results. What is a measurement at the end? is a comparison of an unknow with a know value(s), wrote the famous former editor of the journal AQUAL.

ema in Mexico has open the oportunity to be acredited as a research laboratory. It means laboratorios that are acredited under ISO17025 but perform non routine analysis. The tests of these laboratories does not correspond to a regulation or norm

Thank you so much everyone of you, your answers are really useful and help me a lot, the diversity in the perspectives enriches a lot the debate but it shows me this topic must be analyzed deeply as many of you detailed in your kind answers.

Your partner is incorrect. Accreditation does not validate methods; it merely checks the laboratory's ability to keep paperwork (and, in rare instances, have technically competent people). The laboratory should "validate" the method by running sufficient quality control samples to verify that the method is producing the intended results, and have sufficient technical skills to assess the data produced. Unfortunately, especially in the case of EMA-accredited laboratories, the accreditation process does not verify either the lab's ability to produce quality data nor assess the laboratory's technical resources.