Find first 3 orders for each customer

I am trying to find the first 3 timestamps for each customer.

Sample data for table customer_orders:

Desired outcome:

My query so far:

SELECT
    customer_id, 
    timestamp
FROM customer_orders
GROUP BY customer_id, timestamp
ORDER BY timestamp ASC
LIMIT 3

LIMIT 3 limits to 3 rows overall. But I want 3 rows per customer.

You can use the window function ROW_NUMBER() to numerate the columns of a given PARTITION (customer_id in your case) within a CTE and in the outer query just filter the n records from this generated column:

WITH j AS (
  SELECT customer_id, timestamp,
    ROW_NUMBER() OVER (PARTITION BY customer_id ORDER BY timestamp
                       RANGE BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW) AS n
  FROM customer_orders
) 
SELECT customer_id, timestamp FROM j
WHERE n <= 3 
ORDER BY customer_id, timestamp

Demo: db<>fiddle