Travel with an UK residence card (EU family member) after Brexit

Indeed, this is one of the many questions that is still unclear.

The liklihood is high, that while valid, this card will remain proof that your wife is family member of a EU citizen.

Based on the German Embassy text below (which may be country specific) a Schengen Visa must be applied for after Brexit and will be issued based on EU rules for a family member of a EU citizen.

This status will not change because of Brexit, assuming that you will remain a EU citizen (i. e. are not a British citizen).

The best advice would be to contact your Embassy and ask if some form of proof can be issued for travel purposes while she is living outside the European Union, which will probably be in the form of a long term multiple entry visa where the 90/180 day rule may apply.

You should ask before travelling to insure that the correct procedure is being followed. Reporting back here with the result would also be a good idea since this question will, no doubt, be asked by others.

Note:
Assume the following is country specific.

From the German Embassy in the United Kingdom Visa Information

• as of 2019-10-08

IMPORTANT NOTICE
In case of an exit of the United Kingdom from the European Union without a withdrawal agreement (“no-deal Brexit”) the rights of British citizens and their family members to freedom of movement within the Union will no longer apply. This also includes the family members of German nationals residing in the UK under freedom of movement rules.

This means that in case of a “no-deal Brexit” any Schengen visa issued to a family member of a British or German citizen (on the basis of EU freedom of movement rules) will cease to be legally valid the moment the UK’s membership in the European Union ends. In such case the visa will therefore no longer be recognized as valid for entry into Germany with immediate effect and irrespective of the validity dates stated on the visa.

This implies that no generous solution is on offer.

Existing visa's will become invalid and new ones must be applied for after Brexit.

Do you need a visa for Germany?

• as of 2019-10-08

You DO NOT need a visa for short stays in Germany if you are:
...
a spouse (married/civil partnership) or child of an EU/EEA/EFTA national must hold one of the following British visas that say explicitly: “Residence Card of a Family Member of an EU Citizen” OR “Residence Card of a Family Member of an EEA National” OR “Permanent Residence Card” - and must travel with the EU/EEA/EFTA national. Please compare displayed samples (British Residence Cards ) carefully; If you intend to travel on your own you must apply for a regular visitor, business or tourist visa. Regular visa fees will apply. Click here for more information on visa fees. Please be aware this [not needing a visa] does not apply to holders of a residence card under the EU Scheme SETTLEMENT OR PRE-SETTLEMENT, please apply for a visa.

Note:
There is no statement on this page that states explicitly that it is valid after Brexit.