eingespannt
busy
Colloquial for very busy or occupied by many tasks and obligations.
Ich kann heute Abend nicht kommen, ich bin beruflich total eingespannt.
I can't come this evening; I'm totally tied up with work.

busy / clamped / harnessed
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busy
Colloquial for very busy or occupied by many tasks and obligations.
Ich kann heute Abend nicht kommen, ich bin beruflich total eingespannt.
I can't come this evening; I'm totally tied up with work.

'Eingespannt' is the past participle of the separable verb 'einspannen' (to clamp in; to harness). The basic meaning is to put something firmly into a device. This idea connects all meanings: you are 'eingespannt' (tied up) with work, a workpiece is 'eingespannt' (clamped) in a vise, and a horse is 'eingespannt' (harnessed) to a carriage.
Der Techniker spannt die Platte in die Maschine ein.
The technician clamps the plate into the machine.
The context reveals the meaning. If people are being discussed, it means 'busy'. If a tool or material is being discussed, it means 'clamped'. If an animal is being discussed, it means 'harnessed'.
Mein Kollege ist momentan sehr eingespannt und hat keine Zeit.
My colleague is very busy at the moment and has no time.
In the meaning 'very busy', 'eingespannt' is mainly used colloquially. It sounds natural in everyday conversations.
Sorry, ich kann am Wochenende nicht helfen, da bin ich privat voll eingespannt.
Sorry, I can't help on the weekend; I'm completely tied up with personal commitments then.
When it means 'busy', 'eingespannt' is often combined with adverbs like 'beruflich' (professionally / for work), 'zeitlich' (time-wise), 'stark' (heavily), or 'total' (totally) to describe the reason or intensity.
Sie ist beruflich stark eingespannt und reist sehr viel.
She is heavily tied up with work and travels a lot.
free
'frei' and 'eingespannt' often form an opposite pair in everyday language when it comes to time and obligations. Someone who 'frei hat' (has the day/time off) does not have to work, go to school, or take care of other fixed tasks at that moment. Someone who is 'eingespannt' has many tasks and little time. 'eingespannt' sounds colloquial and fits especially well when someone is very busy because of work, appointments, or family.

Vor der Reise ist das Team so eingespannt, dass niemand Zeit für eine Pause hat.
Before the trip, the team is so tied up that no one has time for a break.
Am Freitag hat Frau Wagner frei und macht einen langen Spaziergang im Park.
Ms. Wagner has Friday off and is taking a long walk in the park.