Short answer
Yes, when the short form is a name that belongs to Allah alone. Ar-Rahman, al-Khaliq, ar-Razzaq and al-Quddus describe only Allah, so calling a person "Rahman" or "Razzaq" is not allowed in the Hanafi school. Names Allah also uses for people, such as Karim, Aziz, Rahim or Hakim, may be used on their own.
Calling a person by a name exclusive to Allah, without Abd before it, is not permissible and is sinful. Use the full name, a respectful nickname or a kunya instead. Names that are not exclusive to Allah may be used on their own.
The four schools agree on this.
Explanation
Names such as Abdur Rahman mean "servant of the Most Merciful". The beauty of the name is in the word Abd (servant). When the Abd is dropped, the person is called by Allah's own name. The four schools agree that this is wrong for the names that belong to Allah alone.
Names that belong only to Allah (keep the Abd): Allah, ar-Rahman, al-Khaliq, ar-Razzaq, al-Quddus, al-Qayyum, as-Samad. So say Abdur Rahman, Abdul Khaliq, Abdur Razzaq, Abdul Quddus or Abdus Samad in full.
Names that can describe people too (may stand alone): Karim, Aziz, Rahim, Ra'uf, Hakim, Ali, Rashid, Hadi, Malik. The Quran calls the Prophet ﷺ kind and merciful (ra'uf, rahim), and calls the minister of Egypt al-Aziz. So Abdul Aziz may be called Aziz, and Abdul Karim may be called Karim, though the full name is still better.
In many South Asian families the father's name becomes a surname, so a child ends up as "Ahmed Rahman" on paper. Many families write the full form, Abdur Rahman or Abdul Rahman, on documents to avoid this. In daily speech, a short form such as Abdu, or a kunya such as Abu Yusuf, keeps the respect without the problem. See also names to avoid and the meanings of Abdur Rahman and Abdur Razzaq.
The evidence
Quran, Surah Al-Israa 17:110
قُلِ ٱدْعُوا۟ ٱللَّهَ أَوِ ٱدْعُوا۟ ٱلرَّحْمَٰنَ ۖ أَيًّۭا مَّا تَدْعُوا۟ فَلَهُ ٱلْأَسْمَآءُ ٱلْحُسْنَىٰ ۚ وَلَا تَجْهَرْ بِصَلَاتِكَ وَلَا تُخَافِتْ بِهَا وَٱبْتَغِ بَيْنَ ذَٰلِكَ سَبِيلًۭا ﴿١١٠﴾
Say, "Call upon Allah or call upon the Most Merciful. Whichever [name] you call - to Him belong the best names." And do not recite [too] loudly in your prayer or [too] quietly but seek between that an [intermediate] way.
Ar-Rahman is used as a proper name of Allah, like the name Allah itself.
Quran, Surah At-Tawba 9:128
لَقَدْ جَآءَكُمْ رَسُولٌۭ مِّنْ أَنفُسِكُمْ عَزِيزٌ عَلَيْهِ مَا عَنِتُّمْ حَرِيصٌ عَلَيْكُم بِٱلْمُؤْمِنِينَ رَءُوفٌۭ رَّحِيمٌۭ ﴿١٢٨﴾
There has certainly come to you a Messenger from among yourselves. Grievous to him is what you suffer; [he is] concerned over you and to the believers is kind and merciful.
Allah describes the Prophet as ra'uf and rahim, so these words can describe people.
Quran, Surah Yusuf 12:30
۞ وَقَالَ نِسْوَةٌۭ فِى ٱلْمَدِينَةِ ٱمْرَأَتُ ٱلْعَزِيزِ تُرَٰوِدُ فَتَىٰهَا عَن نَّفْسِهِۦ ۖ قَدْ شَغَفَهَا حُبًّا ۖ إِنَّا لَنَرَىٰهَا فِى ضَلَٰلٍۢ مُّبِينٍۢ ﴿٣٠﴾
And women in the city said, "The wife of al-'Azeez is seeking to seduce her slave boy; he has impassioned her with love. Indeed, we see her [to be] in clear error."
Al-Aziz is used for a man, so Aziz is not exclusive to Allah.
Sahih Muslim 2132
Sahih... إن أحب أسمائكم إلى الله عبد الله وعبد الرحمن ...
"The most beloved of your names to Allah are Abdullah and Abdur Rahman."
The honour of these names lies in servanthood to Allah.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Hazr wal-Ibaha, discussion of names. Discusses names that may not be given to people.
- al-Fatawa al-Hindiyya, Kitab al-Karahiyya, chapter on naming children and their kunyas. On names of Allah used for people.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.