MySQL slave stuck at 'System lock' most of the time, high CPU usage

Let me describe what we are facing now:

We have a MySQL Master-Slave, Row Based Replication setup and at the moment, the slave is stuck with a perpetual System Lock. The Master is an active server with many updates and deletes running, but the slave will just not replicate anything. There are no errors in the log and the slave is reading binary logs fine from the master, but it does nothing on the slave. The Seconds_Behind_Master value keeps increasing. This is what show processlist on slave shows:

mysql> show processlist ;
+----+-------------+-----------+------+---------+-------+----------------------------------+------------------+
| Id | User        | Host      | db   | Command | Time  | State                            | Info             |
+----+-------------+-----------+------+---------+-------+----------------------------------+------------------+
| 10 | system user |           | NULL | Connect |  4985 | Waiting for master to send event | NULL             |
| 11 | system user |           | NULL | Connect | 53715 | System lock                      | NULL             |
| 14 | root        | localhost | NULL | Sleep   |  2958 |                                  | NULL             |
| 16 | root        | localhost | NULL | Query   |     0 | init                             | show processlist |
+----+-------------+-----------+------+---------+-------+----------------------------------+------------------+
4 rows in set (0.00 sec)

And show slave status shows:

SQL_Remaining_Delay: NULL
Slave_SQL_Running_State: System lock

The only sign of life is that Relay_Log_Pos from the slave status is changing values but very slowly. Does it mean that it is running the queries from the binary log, but just that it is too slow?

Pasting some info from show engine innodb status :

LIST OF TRANSACTIONS FOR EACH SESSION:
---TRANSACTION 0, not started
MySQL thread id 14, OS thread handle 0x7f7824698700, query id 216193 localhost root init
show engine innodb status
---TRANSACTION 230426904, not started
mysql tables in use 1638, locked 1638
MySQL thread id 11, OS thread handle 0x7f7824920700, query id 216192 System lock
--------
FILE I/O
--------
I/O thread 0 state: waiting for completed aio requests (insert buffer thread)
I/O thread 1 state: waiting for completed aio requests (log thread)
I/O thread 2 state: waiting for completed aio requests (read thread)
I/O thread 3 state: waiting for completed aio requests (read thread)
I/O thread 4 state: waiting for completed aio requests (read thread)

Also noticed that the mysqld process for this particular mysql instance has high CPU usage. What could be causing the Slave_SQL_Running_State: System lock and blocking the slave from running the queries? There are no disk issues with the machine and there is another MySQL instance running which is a slave to another MySQL instance (myisam) on the same master and that is running fine.

MySQL version : 5.6.20. OS : RHEL 6.5 We have some tables that are partitioned (if that matters).

Edit: Just noticed that Slave_SQL_Running_State is occasionally changing to 'Reading event from the relay log'. So it looks like the replication is just too slow.