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Yes, it is, they are both independent.<br />
Domain functional level (Older Windows Server version in your Domain controllers)<br />
Forest Functional level (Older Windows Server version in your whole infrastructure)
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You should clarify what your needs are,<br />
Basically, you do not have older DCs, 2008 and higher, but you do not want to enable the newer ad features?<br />
Group policy preferences and other feature that would be one active with the higher forest level?
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Is it possible to &nbsp;raise the Domain Functional level to Windows 2008 by keeping Forest level &nbsp;as 2003 .
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Is it possible to  raise the Domain Functional level to Windows 2008 by keeping Forest level  as 2003 .

Yes, it is, they are both independent.
Domain functional level (Older Windows Server version in your Domain controllers)
Forest Functional level (Older Windows Server version in your whole infrastructure)

You should clarify what your needs are,
Basically, you do not have older DCs, 2008 and higher, but you do not want to enable the newer ad features?
Group policy preferences and other feature that would be one active with the higher forest level?

Active Directory Functional Level

This topic area includes legacy versions of Windows prior to Windows 2000: Windows 3/3.1, Windows 95 and Windows 98, plus any other Windows-related versions including Windows Mobile.

Windows OS

Windows Server 2008 and Windows Server 2008 R2, based on the Microsoft Vista codebase, is the last 32-bit server operating system released by Microsoft. It has a number of versions, including including Foundation, Standard, Enterprise, Datacenter, Web, HPC Server, Itanium and Storage; new features included server core installation and Hyper-V.

Windows Server 2008

Active Directory (AD) is a Microsoft brand for identity-related capabilities. In the on-premises world, Windows Server AD provides a set of identity capabilities and services, and is hugely popular (88% of Fortune 1000 and 95% of enterprises use AD). This topic includes all things Active Directory including DNS, Group Policy, DFS, troubleshooting, ADFS, and all other topics under the Microsoft AD and identity umbrella.

Active Directory

Windows Server 2003 was based on Windows XP and was released in four editions: Web, Standard, Enterprise and Datacenter. It also had derivative versions for clusters, storage and Microsoft’s Small Business Server. Important upgrades included integrating Internet Information Services (IIS), improvements to Active Directory (AD) and Group Policy (GP), and the migration to Automated System Recovery (ASR).