Yes, jīv can exist in pen ink, but how much and what kind depends on the type of ink.

In Jain Darshan, wherever there is:

• water / moisture
• organic (carbon-based) material
• and suitable temperature

there is a strong possibility of sūkṣma jīv (very tiny living beings), especially water-bodied (ap-kāy) jīv and nigod.

1. Traditional / water-based inks

Most ordinary writing inks today are:

• water-based (or partly water + solvents)
• mixed with dyes, gums, etc.

Because of the water part, such ink can contain countless micro-jīv, just like normal water. So from a Jain point of view, we assume there are jīv present.

2. Very dry / solvent-heavy inks

• Solid ink sticks, completely dry powder, or some strong chemical/solvent inks have less scope for jīv, but:

- As soon as moisture / water mixes (air humidity, mixing, etc.), jīv can again arise.

• So practically, we do not claim “no jīv at all” in any such material.

3. Practical conduct (vyavahār)

• For sādhus–sādhvīs, any use of liquids, colors, writing etc. is done with very strict rules to avoid unnecessary himsā.
• For śrāvaks (laypeople), using pen ink for study, work, etc. is treated as necessary worldly activity.

- We still try to minimize himsā by: - avoiding waste - not playing unnecessarily with ink / colors - keeping general ahimsā-bhāv.

So, in simple words:

• Yes, in most liquid pen ink, especially water-based ink, Jain view will consider that jīv can be present.
• But normal, careful use of a pen for writing is accepted for householders as part of vyavahār, while maintaining as much ahimsā and simplicity as possible.

You can read more about life in water and such subtle jīv here