There were indeed other prophets around the time of Muhammad. Particularly, there were a series of self-declared prophets who arose (and were swiftly defeated) during the Wars of Ridda (Apostasy) that occurred after the death of the prophet Muhammad in 632 AD.

Many tribes who had previously converted to Islam and pledged allegiance to Muhammad reneged both upon Islam and upon paying various taxes (for example the Islamic zakat tax) claiming that they had entered into an agreement with Muhammad, and that it was void after his death. These rebellions then tended to center themselves around a self-declared prophet who would lead the fight against the Meccan Muslims. Abu Bakr, the first Caliph to succeed Muhammad, was committed to destroying these rebellions and reinforcing the power and necessity of Islam in the Arabian Peninsula. So the Muslims were aware of these prophets and actively fought against them, causing many of their teachings to die out.

Just a few of the main prophets were:

Musaylimah - Also referred to as al-Kādhib (the Liar), Musaylimah was an Arab of Christian origin from the tribe of the Banū Ḥanīfa. Due to his background, his teachings had Christian leanings such as an emphasis on the Kingdom of Heaven. Other aspects were shared with Islam, such as daily prayers (3 instead of 5) and an aversion to pork. Musaylimah was defeated in the Battle of Yamama in late 632. However, the battle was large enough and bloody enough (the walled garden where the last stand of Musaylimah took place was called the Garden of Death) that it resulted in the deaths of many Qur'anic reciters, one of the reasons for Abu Bakr to start compiling a written copy of the Qur'an.

Sajah - Was another Christian, this time a woman of the Banū Tamīm. I can't find much on her teachings themselves, but she led a force of around 4,000 rebels and later joined with Musaylimah. Some say she married him, others that they merely came to an agreement to work together. Following Musaylimah's defeat, Sajah returned to her tribe, converted to Islam, and lived a rather normal life.

Tulayha - was a Tribal Chief and kāhin, or soothsayer, who led another failed rebellion against the nascent Muslim state. Being a kāhin he would have dealt with divinations, communications with djinn, and other pre-Islamic polytheistic rituals. Notably, Tulayha started his campaign before Muhammad died (different from Musaylimah or Sajah) however much of the impetus was the same. Many of the tribes who came to support Tulayha did so to reduce their tax burden and gain more independence, not because of strict religious belief. Tulayha and his coalition were defeated in 632 and he officially repented and converted to Islam in 634, becoming an important member of the Islamic armies advancing North to fight against Byzantium and Persia. It is fighting the Persians that he did in 642.

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So to sum up:

Yes, there were other prophets around the time of Muhammad. Some of them were linked to the Christian Abrahamic tradition and others to pre-Islamic polytheistic religions. Either way, they gathered a substantial number of followers, enough to threaten the emerging Islamic Empire, and were suppressed during the Wars of Ridda. However, not all of their followers were religious devotees; many fought for political reasons and to reassert the authority they had ceded to Muhammad.

Someone mentioned the copycat prophets that sprung up, Musaylima, Sajah and Tulayha so I'll add the ancient prophets.

Islamically, the only Arabian prophets mentioned in the Qur'an are Hud and and Saleh who were sent to the ancient civilizations of 'Ad and Thamud which existed in southern arabia. They were descended of Noah.The Islamic narrative is that like all prophets they preached monotheism, and were Muslims. Both of the civilizations rejected the message and were destroyed, you can see their ruins today.