SBS2011 can not ping itself by name

Run a general reset on TCP/IP and DNS Flush

Open cmd.exe with Run as Administrator
Then: netsh int ip reset c:\resetlog.txt
Then: ipconfig /flushdns

Then: restart the server

C:\Windows\system32>netsh int ip reset c:\resetlog.txt
Reseting Global, OK!
Reseting Interface, OK!
Reseting Unicast Address, OK!
Reseting Route, OK!
Restart the computer to complete this action.
C:\Windows\system32>ipconfig /flushdns
Windows IP Configuration
Successfully flushed the DNS Resolver Cache.

I rebooted and will report back.

Remember computer name + DNS lookup of a host or domain are 2x completely different actions.

You can never ping a computer name, unless the computer name exactly matches a host or domain.

Also, 192.168.1.9 is in the private network address space, so you can only ping this IP inside your local network, as where ever you ping this address, the response will only come from inside your local network.

Tip: Use nslookup to determine if your "computer name" resolves + if so, what IP is returned.

@John
it did not resolve the problem.

@David,
nslookup failed when I looked up by computer name or ip address 192.168.1.1

Here are details of error when NSLOOKUP is run:
C:\Windows\system32>nslookup p1
DNS request timed out.
    timeout was 2 seconds.
Server:  UnKnown
Address:  fe80::b014:f5f3:9029:7ef5

DNS request timed out.
    timeout was 2 seconds.
DNS request timed out.
    timeout was 2 seconds.
*** Request to UnKnown timed-out

Do you need to ping by name?

This Microsoft Article presents a possible solution for you.
https://social.technet.microsoft.com/Forums/ie/en-US/f82b747e-6711-418b-9b6f-317223daa2dc/access-computers-by-ip-addresses-but-not-by-computer-name?forum=w7itpronetworking

I just logged on a client 2012  R2 server and you cannot ping it by its standard name.  You would need to get the full name of the server

Do you need to ping by name? --> If everything is in working order, yes it should.
Due to this problem, I think, I can't surf the internet from this server.
From backup domain controller (192.168.1.3), I can ping computers by IP or name. I can run NSLOOKUP too.

Pinging by name or IP does not affect ability to surf the internet. I can do that just fine.

"Pinging by name or IP does not affect ability to surf the internet." --> Yes. In fact, if I replace primary name server to 192.168.1.1 which is default gateway, I can surf the internet. Or if I enter backup domain controller IP address 192.168.1.3 as secondary name server, I can surf the internet that way too.

The problem is that it is fundamental issue. Due to this issue, there are other things that are not working.

I am working on a Server 2012 R2, all functions including internet and browsing are working . I can ping the name from outside just fine but DNS is resolving to IP address.

So I don't think your symptom is the cause

I ran Active Directory Replication Status Tool and it fails on "get Domain Controller Replication Status" on both P1(Primary DC) and P2(Backup DC).
The strange thing is that the computer entries in DNS manager on both P1 and P2 are identical.

I am not sure.  I looked at David's advice above and it seems to fit my situation.

One thing I might try if this were me would be to restart DHCP services and the restart DNS services

https://www.oreilly.com/library/view/windows-server-cookbook/0596006330/ch13s03.html

If "nslookup failed when I looked up by computer name", you have 2x general choices.

1) Arrange for the name you're using to resolve to an IP. You'll do this via a DNS record or /etc/hosts equivalent.

2) Ping by IP, rather than name.

I suspect this DNS issue is contributing to the DHCP problem too.
We are unable to manage DHCP. When I run DHCP MMC, it shows a red circle.
Another problem is I can't update group policy.

There are some ideas here.  The person did a system restore.

https://community.spiceworks.com/topic/1999789-sbs-2011-dns-or-ad-issue

can you restore to a prior point?

Even netdom query fails:

C:\Windows\system32>netdom query fsmo
Access is denied.

I attached the result of dcdiag /v and below is error listing:
 
(1) Directory Binding Error 5:
(2)   * Active Directory LDAP Services Check
         The host 19967ded-02c6-46b6-8096-77ed080bcbb3._msdcs.PK.local
         could not be resolved to an IP address. Check the DNS server, DHCP,
         server name, etc.
 Got error while checking LDAP and RPC connectivity. Please check your
         firewall settings.
(3) Testing server: Default-First-Site-Name\PK1
      Skipping all tests, because server PK1 is not responding to
      directory service requests.
(4)   Running enterprise tests on : PK.local
      Test omitted by user request: DNS
      Test omitted by user request: DNS
      Starting test: LocatorCheck
         GC Name: \\PK1.PK.local
         Locator Flags: 0xe00033fd
         [PK1] DsBindWithSpnEx() failed with error 1722,
         The RPC server is unavailable..
         Printing RPC Extended Error Info:
         Error Record 1, ProcessID is 4176
          (DcDiag)

(5)
            Status is 1722 The RPC server is unavailable.
DCDiagnosis.txt

The answer here in a Microsoft thread suggests calling Microsoft Support.  This be a paid call but may be worth it to get your server running

https://social.technet.microsoft.com/Forums/en-US/3c69f804-43c4-45f3-90f2-628ce517fb78/sbs-2011-dns-server-not-responding?forum=smallbusinessserver

Maybe DNS server database is corrupted?
Can I repair DNS database, if it exists or restore from the backup?

Do you have a backup?

https://servergurunow.wordpress.com/2017/09/26/dns-backup/

@John,
 
 https://servergurunow.wordpress.com/2017/09/26/dns-backup/ --> It looks to complicated to try.

If nothing works out in the next couple of days, I will use "paid service" from microsoft.

But before I resort to that, can I restore all the files (from C:\Windows\System32\dns)  on the backup device to the server?
Is that an option?

It is possible - I am not certain. But keep backups of all you do.

Did you ping by IPv4 address

ping P1 -4

Select allOpen in new window

I fixed the problem by choosing “Recommended Setting” in Windows Firewall.

Thank you all!

Thanks for the update. Hard to know what networking device, but for sure not just the pinging symptom