In the US they're called "personal days", and I think it's also gaining usage in the UK.

"As an employee, it is important that you make use of the benefits provided to you by your company. Some of these benefits may include personal time off or personal days.
...
What are personal days at work?
Personal days are a form of employee benefit that employers offer in addition to sick days, PTO and vacation days."
https://www.indeed.com/career-advice/career-development/pers...

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Note added at 15 mins (2021-07-25 18:50:08 GMT)
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"HR Guide: Personal Days - North Ayrshire Councilhttps://www.north-ayrshire.gov.uk › Documents
PDF
Personal Days allow an employee to take up to three days of their annual ... Line manager emails [email protected] advising of ..."

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Note added at 34 mins (2021-07-25 19:10:00 GMT)
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Here's the relevant text from the North Ayrshire Council document I was trying to link to:

"Personal Days allow an employee to take up to three days of their annual leave entitlement (per leave year) at short notice. The needs of the service and the possible costs associated with cover arrangements must be considered by the manager prior to authorisation and there is no guarantee that requests can be accommodated. Employees do not require to provide details of the reason for the request.
Examples of Personal Days
• Unplanned personal or private matters
• Illness of dependent (e.g. whole day is needed to provide care and the paid time off is not covered by Special Leave)
• Feeling under the weather and preferring not to come to work
• Emergency situation (e.g. car/boiler breakdown where employee wants to make repair arrangements
• Severe weather conditions where employee prefers not to travel (out with the Adverse Weather Policy)
Personal days are not an additional entitlement to annual leave but a way of requesting leave at short notice. Personal days are deducted from the employee’s annual leave entitlement as normal."

https://www.opm.gov/policy-data-oversight/pay-leave/leave-ad...

That's what they used to be called at offices in the Greater London Area, 'spares' at our school in the South of England being shorthand for a 'spare or no lesson'.

There's a reference about extra days holiday time granted by employers. "This can be relevant to part time workers if they work bank holidays and their employer can allow them another day off later, if the employer decides to do so. At the employer's discretion, workers can have an extra half-day off if the employer decides to close early on Christmas Eve, for instance. Thus, extra days off for part time workers seem to depend on an employer's discretion."

(SP: Hay un referente sobre días libres del lugar del trabajo que son otorgados por el jefe o el gerente. Pueden tener que ver con trabajadores de horarios parciales si trabajan días de feria. A discreción o a criterio del empleador, trabajadores pueden tener un medio día libre si el empleador se decide a cerrar temprano el día antes de Navidad. Días libres discrecionales para trabajadores dependen de los decisos del jefe.)

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Note added at 2 days 10 hrs (2021-07-28 05:12:48 GMT) Post-grading
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I understood that days off work could be either paid or unpaid by the employer. In the past, legal employers would be obliged to pay for only two days off work due to illness per year. However allowances could increase with increasing length of service in the same place of work, or in the same line of employment I expect that time off for any other reason, paid or unpaid, would need to be agreed by the employer.

(ESP: Entendía que los días fuera del lugar de trabajo podían ser pagados o no pagados. En el pasado, empleadores legales serían obligados a pagar sólo dos días de descanso a casa debidos a enfermedad cada año. Sin embargo, subsidios podían incrementarse con la duración de la duración del servicio en el mismo lugar de trabajo, o en el mismo tipo de trabajo. Supongo que días libres de trabajo, pagados o no pagados, tendrían que ser aprobados por los patrones.)

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Note added at 2 days 10 hrs (2021-07-28 05:15:42 GMT) Post-grading
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I wrote 'el día antes de Navidades' as 'the day before Christmas'. I understood 'la noche buena' to be Christmas Eve at night.

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Note added at 4 days (2021-07-30 13:28:10 GMT) Post-grading
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Holiday pay is similar with legal employers, who are obliged to pay workers for a set number of paid holidays each year. Paid holiday time can increase with the length of continuous service. Unpaid holiday time can take place when employers are not legally obliged to include holiday pay.

(ESP: Vacaciones pagadas son similares, puesto que los días de vacaciones que los patrones de trabajo están obligados a pagar cada año incrementan con la duración del servicio continuo. Vacaciones no pagadas tienen lugar cuando no están obligados a pagar las vacaciones.)